The Chain Rule
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.
The Chain Rule for Nested Functions
Differentiate outside to inside, multiply the derivatives
Illustration: chain-rule-nested
Chain Rule
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
Practice
1. d/dx[e^(2x)] = ?
Chain rule: e^(2x) * 2 = 2e^(2x)
2. d/dx[sqrt(x^2 + 1)] = ?
(1/2)(x^2+1)^(-1/2) * 2x = x/sqrt(x^2+1)
Key Takeaways
Chain Rule: [f(g(x))]' = f'(g(x)) * g'(x)
"Derivative of outside times derivative of inside"
The most frequently used rule in calculus
Don't forget to multiply by the inner derivative!