# Section 2.5

## Table of Contents

The chain rule

# The Chain Rule #

My motto, which you’ll hear me say again and again: “it’s always the chain rule.”

We can differentiate polynomials, trigonometric functions, rational (fractional) functions, etc… but what about functions that are compositions of multiple functions?

## The chain rule: #

If $f$ and $g$ are differentiable functions and $y = f(g(x))$, then:

$$ \frac{d}{dx} (f(g(x)) = f’(g(x)) g’(x) $$

### Demo example #

Find the derivative of $(3x+1)^2$ in two ways. First by using the chain rule, then by expanding it and differentiating the polynomial.

### Examples #

- Find the derivative of $y = \sin(x^2)$
- Find the derivative of $y=\sin^2 (x)$

- Find $\dfrac{d}{dx} \sqrt{3x^2 - 4x}$

- Find $\dfrac{dy}{dx}$ where $\displaystyle y = \left(\frac{t}{1-2t}\right)^3$

## Super examples! #

- Find $k’(t)$ if $k(x) = \sin(\cot (\sec x))$.

- Find the derivative of $f(x) = \sin^2(\tan (x^2))$

# Now practice! #

Head over to WebAssign and work on the exercises.

Have a question? A comment? Want to clarify something above or on your homework set? Let me know!