# Product Rule

This discussion will focus on the Product Rule of Differentiation. This rule states that:

The derivative of the product of two functions is equal to the first function multiplied by the derivative of the second function plus the second function multiplied by the derivative of the first function.

PRODUCT RULE OF DIFFERENTIATION:

$\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$

Let's work some examples:

To work these examples requires the use of various differentiation rules. If you are not familiar with a rule go to the associated topic for a review.

2x(ex)
 Step 1: Simplify the expression This expression is already simplified. 2x(ex) Step 2: Apply the product rule. $\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$ $\frac{d}{dx}\left[2x\left({e}^{x}\right)\right]$ $2x\frac{d}{dx}{e}^{x}+{e}^{x}\frac{d}{dx}2x$ Step 3: Take the derivative of each part. Use the natural exponential rule (NER) to differentiate ex Use the constant multiple and power rules (CM/P) to differentiate 2x. $\frac{d}{dx}{e}^{x}={e}^{x}$    NER _________________________ $\frac{d}{dx}2x=2\frac{d}{dx}x={2}$    CM/P Step 4: Substitute the derivatives into the product rule & simplify. $2x\left({e}^{x}\right)+{e}^{x}\left({2}\right)$ $2{e}^{x}\left(x+1\right)$
Example 1:      x2(6 + 9x)
 Step 1: Apply the product rule. $\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$ $\frac{d}{dx}\left[{x}^{2}\left(6+9x\right)\right]$ ${x}^{2}\frac{d}{dx}\left[\left(6+9x\right)\right]+\left(6+9x\right)\frac{d}{dx}{x}^{2}$ Step 2: Take the derivative of each part. To differentiate 6 + 9x apply the sum rule, the constant multiple rule and then the constant and power rules. To differentiate x2apply the power rule. $\frac{d}{dx}6+9x$ $\frac{d}{dx}6+\frac{d}{dx}9x$ Sum rule $\frac{d}{dx}6+9\frac{d}{dx}x$ CM $0+9={9}$     C&P _____________________ $\frac{d}{dx}{x}^{2}={2}{x}$ Step 3: Substitute the derivatives & simplify. ${x}^{2}\left({9}\right)+\left(6+9x\right)\left(2x\right)$ $9{x}^{2}+12x+18{x}^{2}$ If the expression is simplified first, the product rule is not needed. Step 1: Simplify first. 6x2 + 9x3 Step 2: Apply the sum rule. $\frac{d}{dx}\left[6{x}^{2}+9{x}^{3}\right]$ $\frac{d}{dx}6{x}^{2}+\frac{d}{dx}9{x}^{3}$ Step 3: Take the derivative of each part. To differentiate 6x2 apply the constant multiple and power rules. To differentiate 9x3 apply the constant multiple and power rules. $\frac{d}{dx}6{x}^{2}$ $6\frac{d}{dx}{x}^{2}$          CM $6\left(2x\right)={12}{x}$ Power ______________________ $\frac{d}{dx}9{x}^{3}$ $9\frac{d}{dx}{x}^{3}$        CM $9\left(3{x}^{2}\right)={27}{x}^{2}$ Power Step 4: Add/Subtract the derivatives & simplify. Write polynomials in descending order. $12x+27{x}^{2}$ $27{x}^{2}+12x$
Example 2:      $\left({x}^{5}+6\right)\left(\frac{1}{2}{x}^{4}-1\right)$
 Step 1: Apply the product rule. $\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)\frac{d}{dx}\left[f\left(x\right)\right]$ $\frac{d}{dx}\left[\left({x}^{5}+6\right)\left(\frac{1}{2}{x}^{4}-1\right)\right]$ $\left({x}^{5}+6\right)\frac{d}{dx}\left[\frac{1}{2}{x}^{4}-1\right]+\left(\frac{1}{2}{x}^{4}-1\right)\frac{d}{dx}\left({x}^{5}+6\right)$ Step 2: Take the derivative of each part. To differentiate $\left(\frac{1}{2}{x}^{4}-1\right)$ apply the difference rule, the constant multiple rule and then the constant and power rules. To differentiate $\left({x}^{5}+6\right)$, apply the sum rule and then the constant and power rules. $\frac{d}{dx}\frac{1}{2}{x}^{4}-1$     Original $\frac{1}{2}\frac{d}{dx}{x}^{4}-\frac{d}{dx}1$ Diff. Rule & CM $\frac{4}{2}{x}^{3}-0={2}{x}^{3}$  Constant & Power _____________________ $\frac{d}{dx}\left({x}^{5}+6\right)$ Original $\frac{d}{dx}{x}^{5}+\frac{d}{dx}6$ Sum Rule $5{x}^{4}$ Constant & Power Step 3: Substitute the derivatives & simplify. $\left({x}^{5}+6\right)\left(2{x}^{3}\right)+\left(\frac{1}{2}{x}^{4}-1\right)\left(5{x}^{4}\right)$ $2{x}^{8}+12{x}^{3}+\frac{5}{2}{x}^{8}-5{x}^{4}$ $4.5{x}^{8}-5{x}^{4}+12{x}^{3}$ If the expression is simplified first, the product rule is not needed. Step 1: Simplify first. Distribute the x2. $\frac{1}{2}{x}^{9}-{x}^{5}+3{x}^{4}-6$ Step 2: Apply the sum/difference rule. $\frac{d}{dx}\frac{1}{2}{x}^{9}-\frac{d}{dx}{x}^{5}+\frac{d}{dx}3{x}^{4}-\frac{d}{dx}6$ Step 3: Take the derivative of each part. To differentiate $\left(\frac{1}{2}{x}^{9}\right)$ apply the constant multiple rule and then the power rule. To differentiate (x5), apply the power rule. To differentiate (3x4) apply the constant multiple rule and then the power rule. To differentiate (6) apply the constant rule. $\frac{d}{dx}\frac{1}{2}{x}^{9}=\frac{1}{2}\frac{d}{dx}{x}^{9}=\frac{1}{2}\left(9{x}^{8}\right)=\frac{9}{2}{x}^{8}={4.5}{x}^{8}$ ___________________ $\frac{d}{dx}{x}^{5}={5}{x}^{4}$ _________________ $\frac{d}{dx}3{x}^{4}=3\frac{d}{dx}{x}^{4}=3\left(4{x}^{3}\right)={12}{x}^{3}$ ___________________ $\frac{d}{dx}6={0}$ Step 4: Add/Subtract the derivatives & simplify. $4.5{x}^{8}-5{x}^{4}+12{x}^{3}$

 Related Links: Math algebra Quotient Rule Chain Rule Calculus Topics

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