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The Product Rule of Derivatives



The Formula

the product rule formula
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When is the Product Rule used?

Answer: There are times when it is necessary to find the derivative of two functions that are being multiplied together, Unfortunately, you can not simply take the derivative of each and then multiply, meaning:
the product rule mistake
For example: If $f(x)=2x$ and $g(x)=x^2$ then it is easy to see that these functions can be combined: $$(2x)(x^2)=2x^3$$ Now we can take the derivative: $$\frac{d}{dx}2x^3=6x^2$$ However, if you found the derivative of each $\frac{d}{dx}(x^2)=2x$ and $\frac{d}{dx}(2x)=2$ and then multiplied, you would get $4x$, which is not correct. What if we had to find the derivative of $(x^2)(sin(x))$? In this case, we can not combine the functions into one like the last example, we need another method to find the derivative. This is where the product rule is used
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How does the Product Rule Work ?

Answer: If you have two functions f(x) and g(x) that are differentiable and have known derivatives then to find the derivative of the product of the two functions: You multiply the function by the derivative of the second function and add the second function times the derivative of the first. As seen below:
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Why Does this work?

See Proof Of Product Rule

Notes

If f(x) and g(x) are differentiable at x then
  • h(x)=f(x)g(x) is also differentiable at x.
Based on the associative property, the product rule can be calculated:
  • $f(x)g'(x)+g(x)f'(x)$

Example

Practice Problems

Apply the product rule to each expression below. Use our derivative of functions table to help.


Problem 1)

$tan(x)ln(x)$ (hint)
Answer
Problem 2)

$e^xsin^{-1}x$ (hint)
Answer
Problem 3)

$2x^5cos(x)$ (hint)
Answer
Problem 4)

$2^xsec(x)$ (hint)
Answer
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What if we have more than 2 functions?

If f(x), g(x) and s(x) are differentiable functions with known derivatives then: $$h(x)= \color{red}{f(x)} \color{blue}{g(x)}\color{green}{s(x)} $$ $$h'(x)=\color{red}{f(x)}\color{blue}{g(x)}\color{green}{s \boldsymbol '(x)} + \color{red}{f(x)}\color{blue}{g\boldsymbol '(x)}\color{green}{s(x)} + \color{red}{f\boldsymbol'(x)}\color{blue}{g(x)}\color{green}{s(x)}$$
Examples
$xln(x)sin(x)$ (hint)
Show Derivative
$x^3e^xtan^{-1}x$ (hint)
Show Derivative