Chain rule

Chain Rule Definition

f(g(x)) = f’(g(x)g’(x)

Tangent Line for g around x = a

l(x) = g(a)+ g’(a)(x - a)

tangent line for f around x = g(a)

m(x) = f(g(a)) + f’g(a))(x-g(a))

Chain rule in words

The derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

When to Use the Chain Rule
ex. sin(3x²) or ln(e^x + 1) or (x² + 1)^5

Use the Chain Rule when differentiating a function composed of one function inside another f(g(x))

Chain Rule for Trigonometric Functions
d/dx sin(g(x))

cos(g(x)×g’(x)

Chain rule for exponential functions
e^g(x)

e^g(x) × g’(x)

Chain rule for Natural logarithms
ln(g(x))

1/g(x) × g’(x)

Chain rule for power of functions
if y = g(x)^n

n(g(x)^n-1 × g’(x)

chain rule and implicit differentiation

Use the chain rule when differentiating implicitly, especially when functions involve both x and y. y² = 2y × dy/dx