> For the complete documentation index, see [llms.txt](https://solomons-mathbook.gitbook.io/calculus-basics/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://solomons-mathbook.gitbook.io/calculus-basics/comment-389389887/comment-390151930.md).

# Chain Rule

> One of the **core principles** in Calculus is the Chain Rule.

![image](https://user-images.githubusercontent.com/14041622/40289431-ce3b8652-5cea-11e8-9aba-c40a24f19bdd.png)

[Refer to Khan academy article: Chain rule](https://www.khanacademy.org/math/ap-calculus-ab/ab-derivative-rules/modal/a/chain-rule-review) [`▶ Proceed to Integral rule of composite functions: U-substitution`](https://github.com/solomonxie/solomonxie.github.io/issues/49#issuecomment-395677669)

It tells us how to differentiate `Composite functions`.

![image](https://user-images.githubusercontent.com/14041622/40225236-178dc84c-5abb-11e8-8bbf-098d8df7d3b9.png)

**It must be composite functions, and it has to have** `inner & outer` **functions, which you could write in form of** `f(g(x))`**.** ![image](https://user-images.githubusercontent.com/14041622/40225120-c6ddf174-5aba-11e8-8b1b-2859c7eeb749.png)

### Common mistakes

* Not recognizing whether a function is composite or not
* Wrong identification of the inner and outer function
* **Forgetting to multiply by the derivative of the inner function**
* Computing `f(g(x))` wrongly:

  ![image](https://user-images.githubusercontent.com/14041622/40225381-7744459a-5abb-11e8-848d-9a28c39db99c.png)

### How to identify Composite functions

> Seems a basic algebra101, but actually a quite tricky one to identify.

[Refer to Khan lecture: Identifying composite functions](https://www.khanacademy.org/math/ap-calculus-ab/ab-derivative-rules/modal/v/recognizing-compositions-of-functions)

The core principle to identify it, is trying to re-write the function into a nested one: `f(g(x))`. If you could do this, it's composite, if not, then it's not one.

### Examples

![image](https://user-images.githubusercontent.com/14041622/40226233-add7f910-5abd-11e8-8522-853ca335a2a1.png) It's a composite function, which the inner is `cos(x)` and outer is `x²`.

![image](https://user-images.githubusercontent.com/14041622/40226318-f2bbe668-5abd-11e8-9f47-405b4476c054.png) It's a composite function, which the inner is `2x³-4x` and outer is `sin(x)`.

![image](https://user-images.githubusercontent.com/14041622/40226383-24b4fe48-5abe-11e8-9cb0-9c0bda203c89.png) It's a composite function, which the inner is `cos(x)` and outer is `√(x)`.

## `Two forms of Chain Rule`

The general form of Chain Rule is like this: ![image](https://user-images.githubusercontent.com/14041622/40289487-313c6d84-5ceb-11e8-85a9-2aee2108aa60.png)

But the Chain Rule has another **more commonly used** form: ![image](https://user-images.githubusercontent.com/14041622/40289493-3ddcdfa6-5ceb-11e8-9771-2d93d210a85b.png)

Their results are exactly the same. It's just some people find the first form makes sense, some more people find the second one does.

### Example

![image](https://user-images.githubusercontent.com/14041622/40268880-38210900-5ba8-11e8-8e45-3691ad3b5163.png) Solve: \[Refer to Symbolab worked example.]\(<https://www.symbolab.com/solver/step-by-step/\frac{d}{dx}\left(sqrt\left(3cos^{3}\left(x\right)\right)\right>))

## `Chain rule for exponential function`

Formula: ![image](https://user-images.githubusercontent.com/14041622/46215093-17a0cb00-c36f-11e8-9bf0-5162ea68f30e.png)

Because: ![image](https://user-images.githubusercontent.com/14041622/46215176-4ae35a00-c36f-11e8-973b-10ead67e8e8c.png)

### Example

![image](https://user-images.githubusercontent.com/14041622/46215200-59ca0c80-c36f-11e8-96f9-17989b088acf.png) Solve:

* Apply the `Log power rule` to simplify the exponential function:

  ![image](https://user-images.githubusercontent.com/14041622/46215265-8bdb6e80-c36f-11e8-9e0c-cb8aa8670b9d.png)
* Differentiate both sides:

  ![image](https://user-images.githubusercontent.com/14041622/46215364-c7763880-c36f-11e8-8233-182d5f7c4888.png)