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# System prepended metadata

title: Chain Rule

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# Chain Rule

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## When Do We Use Chain Rule?

We use *chain rule* when we want to take the derivative of a *composite function* (a function that has something other than "$x$" where you would expect to see the "$x$").

**Examples of Composite Functions**

* $f(x) = e^{\sin(x)}$
* $g(x) = (x^2 + 3x)^{100}
* $h(x) = \cos(5x + 3)$

## Basic Form of Chain Rule

**Chain Rule (Prime Notation):** If $f(x) = A(B(x))$, then $f'(x) = A'(B(x)) B'(x)$.

**Chain Rule (Leibnitz Notation):** If $y$ depends on $u$ and $u$ depends on $x$, then $$\displaystyle\frac{dy}{dx} = \displaystyle\frac{dy}{du} \displaystyle\frac{du}{dx}$$. 


## Be Careful To Distinguish Chain Rule from Product Rule

<iframe width="560" height="315" src="https://www.youtube.com/embed/_n7JsnrcfVo" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

## Practice

https://www.khanacademy.org/math/ap-calculus-ab/ab-differentiation-2-new/ab-3-1a/v/common-chain-rule-misunderstandings

## Some More Examples

**Example 1:** 

<iframe width="560" height="315" src="https://www.youtube.com/embed/zqSOYsRQJiE" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

**Example 2:**

<iframe width="560" height="315" src="https://www.youtube.com/embed/xBLDqG0YcbQ" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

**Example 3:** 

<iframe width="560" height="315" src="https://www.youtube.com/embed/qbIb4R16WDc" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>





