Let's explore how to find the derivative of any polynomial using the power rule and additional properties. The derivative of a constant is always 0, and we can pull out a scalar constant when taking the derivative. Furthermore, the derivative of a sum of two functions is simply the sum of their derivatives. Created by Sal Khan.Basic CalculusThe Product Rule for Derivatives | Basic Rules of DerivativesThis video will demonstrate how to find the derivatives of a function using the pr...Feb 15, 2021 ... In other words, it helps to take the derivative of a variable raised to a power (exponent). The Steps. All we have to do is: Move the exponent ...Now use the product rule to determine the partial derivatives of the following function: To illustrate the quotient rule, first redefine the rule using partial differentiation notation: ... Then the rule for taking the derivative is: Use the power rule on the following function to find the two partial derivatives: ...Product rule derivative formula is a rule in differential calculus that we use to find the derivative of product of two or more functions. Suppose two functions , u(x) and v(x) are differentiable , then the product rule can be applied to find (d/dx)[u(x)v(x)] as, This Product Rule Review page, located in the Derivative Rules unit, has examples and exercises that assume knowledge of how to find derivatives of exponential and logarithmic functions. However, those derivatives are not covered …Product rule calculator is an online tool which helps you to find the derivatives of the products. You can evaluate derivatives of products of two or more functions using this product rule derivative calculator. The product rule solver allows you to find products of derivative functions quickly because manual calculation can be long and tricky. Product Rule Example 1: y = x 3 ln x. The derivative of x 3 is 3x 2, but when x 3 is multiplied by another function—in this case a natural log (ln x), the process gets a little more complicated.. Step 1: Name the first function “f” and the second function “g.”Go in order (i.e. call the first function “f” and the second “g”). f = x 3; g = ln xProduct Rule Example Questions. Question 1: Using the product rule, show that the function y = x^3 y = x3 has derivative \dfrac {dy} {dx} = 3x^2 dxdy = 3x2. [2 marks] A Level AQA Edexcel OCR. Question 2: For f (x) = 2\sin x \cos x f (x) = 2sinxcosx, use the product rule to find its derivative with respect to x x, and prove that 2\sin x \cos x ...Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. So, you start with d/dx [ (x^2+1)^3 ] = 3 (x^2+1)^2 (2x) = 6x (x^2+1)^2 (Chain Rule!) Now, do that same type of process for the derivative of the second multiplied by the first factor.While this is certainly not as awful as the quotient rule, it is not as simple as the rule for sums, which was the good-sounding slogan that the derivative of ...We already know that the product rule tells us that if we have the product of two functions-- so let's say f of x and g of x-- and we want to take the derivative of this business, that this is just going to be equal to the derivative of the first function, f prime of x, times the second function, times g of x, plus the first function, so not even taking its derivative, so plus f of …Feb 15, 2021 · Use Product Rule To Find The Instantaneous Rate Of Change. So, all we did was rewrite the first function and multiply it by the derivative of the second and then add the product of the second function and the derivative of the first. And lastly, we found the derivative at the point x = 1 to be 86. Now for the two previous examples, we had ... Collectively the second, third, fourth, etc. derivatives are called higher order derivatives. Let’s take a look at some examples of higher order derivatives. Example 1 Find the first four derivatives for each of the following. R(t) = 3t2+8t1 2 +et R ( t) = 3 t 2 + 8 t 1 2 + e t. y = cosx y = cos.Note: You may know that $\displaystyle\left(\frac 1 h \right)' = \frac {-h'}{h^2}$ could be calculated by product rule, as if one consider the product $\displaystyle\left(\frac 1 h \cdot h \right) = 1$, and calculate the derivative of both sides of the equation. one the left hand side we have a constant which may already know the derivative is $0$, but on the …Seminars are an essential tool for businesses and organizations to share knowledge, educate employees, and connect with their target audience. As seminar organizers, it is crucial ...Understanding the "Chase 5/24 Rule" is key in earning travel rewards. We'll list the cards that are subject to the rule and how to avoid it. We may be compensated when you click on...3 Rules for Finding Derivatives. 1. The Power Rule; 2. Linearity of the Derivative; 3. The Product Rule; 4. The Quotient Rule; 5. The Chain Rule; 4 Transcendental Functions. 1. Trigonometric Functions; 2. The Derivative of $\sin x$ 3. A hard limit; 4. The Derivative of $\sin x$, continued; 5. Derivatives of the Trigonometric Functions; 6 ...The product rule is the method used to differentiate the product of two functions, that's two functions being multiplied by one another . For instance, if we were given the function …Product rule. I would take the derivative of the first expression. So, X, derivative of X squared is two X. Let me write a little bit to the right. This is gonna be two X times the second expression sin of X. Plus the first expression X squared times the derivative of the second one. Cosin of X. The product rule allows us to find the derivative of two functions’ product using the respective functions’ corresponding derivatives. This article will show how we can easily …You don't have to be careful about this when doing the product rule, but when doing the quotient rule, remember that you subtract term with the derivative of the bottom function, and divide by the bottom function squared. Anything else shouldn't give you the right answer, and (e^x (2-x)) / x^3 would be incorrect.Solve derivatives using the product rule method step-by-step with this online calculator. Enter a function and get the derivative of its product, quotient, or sum with respect to any …2. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g' (f ⋅g)′ = f ′⋅ g+f ⋅g′, where f=3x+2 f = 3x+2 and g=x^2-1 g = x2 −1. 3. The derivative of a sum of two or more functions is the sum of the derivatives of each function. 4. The derivative of a sum of two or more functions is the sum of the derivatives of ...Instead, the derivatives have to be calculated manually step by step. The rules of differentiation (product rule, quotient rule, chain rule, …) have been implemented in JavaScript code. There is also a table of derivative functions for the trigonometric functions and the square root, logarithm and exponential function. The director's biggest inspiration for the sequence were the helicopters in "Apocalypse Now." After six seasons of build up over the fearsome power of the dragons, fire finally rai...This behavior illustrates the fact that one can ignore Δ A 3 (the cyan rectangle), when calculating the derivative of A. Since d A 1 d t = d x d t y and d A 2 d t = x d y d t, the applet illustrates the product rule. d A d t = d d t ( x y) = d x d t y + x d y d t. More information about applet. The product rule is motivated by calculating the ...We already know that the product rule tells us that if we have the product of two functions-- so let's say f of x and g of x-- and we want to take the derivative of this business, that this is just going to be equal to the derivative of the first function, f prime of x, times the second function, times g of x, plus the first function, so not even taking its derivative, so plus f of …Jan 21, 2019 · If our function was the product of four functions, the derivative would be the sum of four products. As you can see, when we take the derivative using product rule, we take the derivative of one function at a time, multiplying by the other two original functions. How I do I prove the Product Rule for derivatives? All we need to do is use the definition of the derivative alongside a simple algebraic trick. First, recall the the the product f g of the functions f and g is defined as (f g)(x) = f (x)g(x). Therefore, it's derivative is. (f g)′(x) = lim h→0 (f g)(x + h) − (f g)(x) h = lim h→0 f (x ...Learn how to use the product rule to calculate the derivative of a product of two or more differentiable functions. See the formula, examples, common mistakes, and applications of this rule in calculus.In calculus, the general Leibniz rule, [1] named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if and are -times differentiable functions, then the product is also -times differentiable and its th derivative is given by. where is the binomial coefficient and denotes the j ... Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for …This Product Rule Review page, located in the Derivative Rules unit, has examples and exercises that assume knowledge of how to find derivatives of exponential and logarithmic functions. However, those derivatives are not covered …Product Rule | Product Rule For Derivatives | Derivative Rules | Differentiation Product RuleHi Students !! Welcome back to our channel. In this video I've e...Jan 7, 2017 · The product rule is one of the fundamental derivative rules in calculus. It shows you how to take the derivative of the product of two functions: f·g. In t... a lot of gadgets and gizmos out thereIf you plan to bring a carry-on bag and personal item with you on a United flight, know the rules and restrictions to plan accordingly. We may be compensated when you click on prod...Product Rule : Example Question #4 ... What is the derivative of: \displaystyle [(2x^2+x)(x^2-1)]? ... Step 2: Find \displaystyle f'(x) and \displaystyle g'(x).VANCOUVER, British Columbia, Dec. 23, 2020 (GLOBE NEWSWIRE) -- Christina Lake Cannabis Corp. (the “Company” or “CLC” or “Christina Lake Cannabis... VANCOUVER, British Columbia, D...Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. So, you start with d/dx [ (x^2+1)^3 ] = 3 (x^2+1)^2 (2x) = 6x (x^2+1)^2 (Chain Rule!) Now, do that same type of process for the derivative of the second multiplied by the first factor. Oct 31, 2017 ... Looking at the coefficient of h we see the product rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x). In other words (a + bh)(c + dh) has linear ...What Is The Product Rule? In product rule calculus, we use the multiplication rule of derivatives when two or more functions are getting multiplied. If we have two functions f(x) and g(x), then the product rule states that: “ f(x) times the derivative of g(x) plus g(x) times the derivative of f(x)” Formula of Product Rule: A three-judge court in The Hague ruled that a European patent for teff lacked “inventiveness.” A legal tussle over who owns teff, Ethiopia’s staple grain, has been quietly settled....Dec 29, 2020 · In the following example, we compute the derivative of a product of functions in two ways to verify that the Product Rule is indeed "right.'' Example 51: Exploring alternate derivative methods Let \(y = (x^2+3x+1)(2x^2-3x+1)\). The product rule is used to find the derivative of any function that is the product of two other functions. The quickest way to remember it is by thinking of the general pattern it follows: “write the product out twice, prime on 1st, prime on 2nd”. Continue studying derivatives. Previous: The power rule for derivatives How to prove the product rule derivative using first principle of derivatives. We will prove the product rule by the first principle of derivatives, the definition of the derivative. In other words, we will prove the next equality holds: \begin {equation*} (fg)' (x) = f' (x)g (x) + f (x)g' (x). \end {equation*} (f g)′(x) =f ′(x)g(x)+f (x)g ...The Product Rule. As parts (b) and (d) of Preview Activity \(\PageIndex{1}\) show, it is not true in general that the derivative of a product of two functions is the product of the derivatives of those functions.In this chapter we introduce Derivatives. We cover the standard derivatives formulas including the product rule, quotient rule and chain rule as well as derivatives of polynomials, roots, trig functions, inverse trig functions, hyperbolic functions, exponential functions and logarithm functions. We also cover implicit differentiation, related rates, …Solution. To apply the Product Rule, we first need to identify the two functions being multiplied, and then find the derivative of each: We can now apply the Product Rule: That’s it. As long as you remember to find the derivative of each function separately (even if just in your head) and then make the correct substitutions in the Product ... Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. So, you start with d/dx [ (x^2+1)^3 ] = 3 (x^2+1)^2 (2x) = 6x (x^2+1)^2 (Chain Rule!) Now, do that same type of process for the derivative of the second multiplied by the first factor. How to prove the product rule derivative using first principle of derivatives. We will prove the product rule by the first principle of derivatives, the definition of the derivative. In other words, we will prove the next equality holds: \begin {equation*} (fg)' (x) = f' (x)g (x) + f (x)g' (x). \end {equation*} (f g)′(x) =f ′(x)g(x)+f (x)g ...Unit 9: Product Rule Lecture 9.1. The product rule gives the derivative of a product of functions in terms of the functions and the deriva-tives of each function. It is also called Leibniz rule named after Gottfried Leibniz, who found it in 1684. It is important because it allows us to di erentiate many more functions. We will be able to ...This calculus video tutorial explains how to find the derivative of trigonometric functions such as sinx, cosx, tanx, secx, cscx, and cotx. It contain examp...What Is The Product Rule Formula? The following image gives the product rule for derivatives. Scroll down the page for more examples and solutions. How To Use The Product Rule? Example: Find f’(x) if f(x) = (6x 3)(7x 4) Solution: Using the Product Rule, we get. Example: Given f(x) = (3x 2 – 1)(x 2 + 5x +2), find the derivative of f(x ... Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. So, you start with d/dx [ (x^2+1)^3 ] = 3 (x^2+1)^2 (2x) = 6x (x^2+1)^2 (Chain Rule!) Now, do that same type of process for the derivative of the second multiplied by the first factor.The Leibniz identity extends the product rule to higher-order derivatives. See also Chain Rule, Derivative, Exponent Laws, Leibniz Identity, Quotient Rule, Related Rates Problem Explore with Wolfram|Alpha. More things to try: product rule Bode plot of s/(1-s) sampling period .02;A court ruling says credit card points, miles and cashback can sometimes be taxable. But don't panic yet. Correction 2/25/21: This article has been updated to reflect that Visa gif...The product rule is one of the fundamental derivative rules in calculus. It shows you how to take the derivative of the product of two functions: f·g. In t...Many calculus students know their derivative rules pretty well yet struggle to apply the right rule in the right situation. To ... if you were asked to differentiate f(x)=(3−8x)(2x−7)), you'd apply the product rule, as f(x) is a product of two functions. Comment Button navigates to signup page (2 votes) Upvote. Button navigates to signup page.Basic CalculusThe Product Rule for Derivatives | Basic Rules of DerivativesThis video will demonstrate how to find the derivatives of a function using the pr...There are Now use the product rule to determine the partial derivatives of the following function: To illustrate the quotient rule, first redefine the rule using partial differentiation notation: ... Then the rule for taking the derivative is: Use the power rule on the following function to find the two partial derivatives: ...3. Derivatives. 3.1 The Definition of the Derivative; 3.2 Interpretation of the Derivative; 3.3 Differentiation Formulas; 3.4 Product and Quotient Rule; 3.5 Derivatives of Trig Functions; 3.6 Derivatives of Exponential and Logarithm Functions; 3.7 Derivatives of Inverse Trig Functions; 3.8 Derivatives of Hyperbolic Functions; 3.9 Chain RuleUse the product rule for finding the derivative of a product of functions. Use the quotient rule for finding the derivative of a quotient of functions. Extend the power rule to …1 Answer. Psykolord1989 . · Jim H. Aug 29, 2014. The product rule for derivatives states that given a function f (x) = g(x)h(x), the derivative of the function is f '(x) = g'(x)h(x) + g(x)h'(x) The product rule is used primarily when the function for which one desires the derivative is blatantly the product of two functions, or when the ...If our function was the product of four functions, the derivative would be the sum of four products. As you can see, when we take the derivative using product rule, we take the derivative of one function at a time, multiplying by the other two original functions.Jan 21, 2019 · If our function was the product of four functions, the derivative would be the sum of four products. As you can see, when we take the derivative using product rule, we take the derivative of one function at a time, multiplying by the other two original functions. Find f′(x) f ′ ( x) Step 1. Identify the factors that make up the function. f(x) = 4x3e−2xcos 6x f ( x) = 4 x 3 e − 2 x cos 6 x. Step 2. Differentiate using the product rule. The parts in blue b l u e are the derivatives of the individual factors. f′(x) = [12x2e−2x cos 6x] +[4x3(−2e−2x) cos 6x] +[4x3e−2x(−6 sin 6x)] = 12x2e ... Recognizing the functions that you can differentiate using the product rule in calculus can be tricky. Working through a few examples will help you recognize when to use the product rule and when to use other rules, like the chain rule. Examples. y = x 3 ln x (Video) y = (x 3 + 7x – 7)(5x + 2) y = x-3 (17 + 3x-3) 6x 2/3 cot x; 1. y = x 3 ln x ... Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. So, you start with d/dx [ (x^2+1)^3 ] = 3 (x^2+1)^2 (2x) = 6x (x^2+1)^2 (Chain Rule!) Now, do that same type of process for the derivative of the second multiplied by the first factor.In this session we apply the main formula to a product of two functions. The result is a rule for writing the derivative of a product in terms of the factors and their derivatives. Lecture Video and Notes Video Excerpts. Clip 1: Introduction of Product and Quotient Rules. Clip 2: Introduction to General Rules for Differentiation. Clip 3 ... Learn how to use the product rule formula to differentiate a product of two functions, such as fg (x) = f (x)g (x) or F (x) = uv. See examples with answers and practice problems to …Note: You may know that $\displaystyle\left(\frac 1 h \right)' = \frac {-h'}{h^2}$ could be calculated by product rule, as if one consider the product $\displaystyle\left(\frac 1 h \cdot h \right) = 1$, and calculate the derivative of both sides of the equation. one the left hand side we have a constant which may already know the derivative is $0$, but on the …Step 3: Substitute the derivatives & simplify. x 2 (9) + (6 + 9 x) (2 x) 9 x 2 + 12 x + 18 x 2. 27 x 2 + 12 x If the expression is simplified first, the product rule is not needed. Step 1: Simplify first. 6x 2 + 9x 3. Step 2: Apply the sum rule. d d x [6 x 2 + 9 x 3] d d x 6 x 2 + d d x 9 x 3. Step 3: Take the derivative of each part. To ...The product rule calculator allow us to take the derivative that we cannot multiply easily or quickly. The product rule solver is a totally free and easily available tool for students, scientists, and engineers. This tool gives error-free results with all possible steps and their calculation details i.e. instructions and graphs etc.Jan 11, 2024 · The derivative of the first factor times the second left alone, plus the first left alone times the derivative of the second. The product rule can extend to a product of several functions; the pattern continues – take the derivative of each factor in turn, multiplied by all the other factors left alone, and add them up: \[\frac{d}{dx}\left( f ... Apr 24, 2022 · The derivative of the first factor times the second left alone, plus the first left alone times the derivative of the second. The product rule can extend to a product of several functions; the pattern continues – take the derivative of each factor in turn, multiplied by all the other factors left alone, and add them up: \[\frac{d}{dx}\left( f ... Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. So, you start with d/dx [ (x^2+1)^3 ] = 3 (x^2+1)^2 (2x) = 6x (x^2+1)^2 (Chain Rule!) Now, do that same type of process for the derivative of the second multiplied by the first factor. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for …In this session we apply the main formula to a product of two functions. The result is a rule for writing the derivative of a product in terms of the factors and their derivatives. Lecture Video and Notes Video Excerpts. Clip 1: Introduction of Product and Quotient Rules. Clip 2: Introduction to General Rules for Differentiation. Clip 3 ... Learn how to use the product rule to differentiate a function of two or more functions in calculus. Find the formula, proof and examples of the product rule for different functions …The Buy American rule guideline has changed. According to the new rule, 75% of the components used to make a product must be made in the US. Wouldn’t you love to land a government ...Many calculus students know their derivative rules pretty well yet struggle to apply the right rule in the right situation. To ... if you were asked to differentiate f(x)=(3−8x)(2x−7)), you'd apply the product rule, as f(x) is a product of two functions. Comment Button navigates to signup page (2 votes) Upvote. Button navigates to signup page.Then the quotient rule tells us that F prime of X is going to be equal to and this is going to look a little bit complicated but once we apply it, you'll hopefully get a little bit more comfortable with it. Its going to be equal to the derivative of the numerator function. U prime of X. Times the denominator function. For this function, both f(x) = c and f(x + h) = c, so we obtain the following result: f′ (x) = lim h → 0 f(x + h) − f(x) h = lim h → 0 c − c h = lim h → 0 0 h = lim h → 00 = 0. The rule for differentiating constant functions is called the constant rule. It states that the derivative of a constant function is zero; that is, since a ...Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative. [ citation needed ] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction — each of which may lead to a simplified expression for …The first one examines the derivative of the product of two functions. Although it might be tempting to assume that the derivative of the product is the product of the derivatives, similar to the sum and difference rules, the product rule does not follow this pattern. To see why we cannot use this pattern, consider the function [latex]f(x)=x^2 ...While f(x)g(x) would be (x+1)x^2, f of g of x would be x^2+1. Continuing on with the same example, the f(x)g(x) derivative with the product rule would give x^2+2x(x+1), and the f of g of x derivative would be 2x. Clearly, not the same thing. Moral of the story: Just use the product rule when there are two functions being multiplied together. Remember your product rule: derivative of the first factor times the second, plus derivative of the second factor times the first. So, you start with d/dx [ (x^2+1)^3 ] = 3 (x^2+1)^2 (2x) = 6x (x^2+1)^2 (Chain Rule!) Now, do that same type of process for the derivative of the second multiplied by the first factor. Summary of the product rule. The product rule is a very useful tool for deriving a product of at least two functions. It is a rule that states that the derivative of a product of two functions is equal to the first function f(x) in its original form multiplied by the derivative of the second function g(x) and then added to the original form of the second function g(x) multiplied by the ...
Example 3.3. 1. This function is not a simple sum or difference of polynomials. It’s a product of polynomials. We can simply multiply it out to find its derivative: h ( x) = ( 4 x 3 − 11) ( x + 3) = 4 x 4 − 11 x + 12 x 3 − 33 h ′ ( x) = 16 x 3 − 11 + 36 x 2. This function is not a simple sum or difference of polynomials.. The cheesecake factory date
For this function, both f(x) = c and f(x + h) = c, so we obtain the following result: f′ (x) = lim h → 0 f(x + h) − f(x) h = lim h → 0 c − c h = lim h → 0 0 h = lim h → 00 = 0. The rule for differentiating constant functions is called the constant rule. It states that the derivative of a constant function is zero; that is, since a ...Find the Derivative Using Product Rule - d/d@VAR f(x)=(3x-5)(2x^3-x^2+1) Step 1. Differentiate using the Product Rule which states that is where and . Step 2. By the Sum Rule, the derivative of with respect to is . Step 3. Evaluate. Tap for more steps... Step 3.1. Since is constant with respect to , the derivative of with respect to is .This behavior illustrates the fact that one can ignore Δ A 3 (the cyan rectangle), when calculating the derivative of A. Since d A 1 d t = d x d t y and d A 2 d t = x d y d t, the applet illustrates the product rule. d A d t = d d t ( x y) = d x d t y + x d y d t. More information about applet. The product rule is motivated by calculating the ...Product Rule. Let and be differentiable at . Then is differentiable at and. We illustrate product rule with the following examples: Example 1: Example 2: Try yourself.The product rule allows us to find the derivative of two functions’ product using the respective functions’ corresponding derivatives. This article will show how we can easily …The U.S. government announced that it will end a requirement for foreign visitors to be vaccinated against COVID-19 on May 11, 2023. We may be compensated when you click on product...Medicine Matters Sharing successes, challenges and daily happenings in the Department of Medicine ARTICLE: Human colon cancer-derived Clostridioides difficile strains drive colonic...2. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g' (f ⋅g)′ = f ′⋅ g+f ⋅g′, where f=3x+2 f = 3x+2 and g=x^2-1 g = x2 −1. 3. The derivative of a sum of two or more functions is the sum of the derivatives of each function. 4. The derivative of a sum of two or more functions is the sum of the derivatives of ...The product rule is one of the fundamental derivative rules in calculus. It shows you how to take the derivative of the product of two functions: f·g. In t...The Product Rule. We'd like to be able to take the derivatives of products of functions whose derivatives we already know. For example f ( x )= ( x -2) ( x -1) is a product of two functions, u ( x )= x -2 and v ( x )= x -1, both of whose derivatives we know to be 1. Wouldn't it be nice if the derivative of a product was the product of the ...The product rule is an essential derivative rule used to find the derivative of a function that can be expressed as a product of two simpler expressions. A great example of this type of function is h ( x) = ( x 3 – 2 x + 1) ( x 3 – 4 x 2 + 1). Without the product rule, our option is to either use the formal definition of derivatives or ... If our function was the product of four functions, the derivative would be the sum of four products. As you can see, when we take the derivative using product rule, we take the derivative of one function at a time, multiplying by the other two original functions.The derivative of the first factor times the second left alone, plus the first left alone times the derivative of the second. The product rule can extend to a product of several functions; the pattern continues – take the derivative of each factor in turn, multiplied by all the other factors left alone, and add them up: \[\frac{d} ....