Chain Rule

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

image

Refer to Khan academy article: Chain rule ▶ Proceed to Integral rule of composite functions: U-substitution

It tells us how to differentiate Composite functions.

image

It must be composite functions, and it has to have inner & outer functions, which you could write in form of f(g(x)).

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:

  

How to identify Composite functions

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

Refer to Khan lecture: Identifying composite 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

It's a composite function, which the inner is cos(x) and outer is x².

It's a composite function, which the inner is 2x³-4x and outer is sin(x).

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:

But the Chain Rule has another more commonly used form:

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

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:

Because:

Example

Solve:

• Apply the Log power rule to simplify the exponential function:

  

• Differentiate both sides: