The Chain Rule

The most important derivative rule

The chain rule handles composite functions - functions inside functions. It's arguably the most important rule because it shows up everywhere.

Chain Rule

d/dx[f(g(x))] = f'(g(x)) * g'(x)

Derivative of outer function (evaluated at inner) times derivative of inner function

Chain Rule

Medium

Find d/dx[sin(x^2)]

1

Identify outer and inner

Outer: sin(u), Inner: u = x^2

2

Derivatives

Outer derivative: cos(u) Inner derivative: 2x

3

Apply chain rule

= cos(x^2) * 2x = 2x*cos(x^2)

Answer: 2x*cos(x^2)

Chain Rule with Power

Medium

Find d/dx[(3x + 1)^5]

1

Identify

Outer: u^5, Inner: u = 3x + 1

2

Apply chain rule

= 5(3x+1)^4 * 3 = 15(3x+1)^4

Answer: 15(3x+1)^4

1. d/dx[e^(2x)] = ?

e^(2x)

2e^(2x)

2xe^(2x)

e^2

Chain rule: e^(2x) * 2 = 2e^(2x)

2. d/dx[sqrt(x^2 + 1)] = ?

x/sqrt(x^2+1)

2x/sqrt(x^2+1)

1/(2sqrt(x^2+1))

2x

(1/2)(x^2+1)^(-1/2) * 2x = x/sqrt(x^2+1)

Key Takeaways