Derivatives of Functions

Chain Rule with Trig Functions

Differentiate f(x)=sin(3x²) using the chain rule: outer'(u)·u' where u=3x².

Example

Apply the chain rule to a trigonometric composite.

highlighted = computed this step

Step 1 — Set up

Start with a composite function.

f(x)=sin⁡(3x2)f(x)= \sin(3x^{2})

Step 2 — Identify inner function

Identify the inner function.

u=3x2u= \hlmath{3x^{2}}

Step 3 — Differentiate outer

Differentiate the outer sine function.

ddusin⁡u=cos⁡(3x2)\frac{d}{du}\sin u= \hlmath{\cos(3x^{2})}

Step 4 — Differentiate inner

Differentiate the inner function.

u′=6xu'= \hlmath{6x}

Step 5 — Multiply the factors

Multiply outer derivative by inner derivative.

f′(x)=cos⁡(3x2)⋅6xf'(x)= \hlmath{\cos(3x^{2})\cdot 6x}
chain-with-trig When a trig function has a composite argument, apply the chain rule: identify the inner function u, differentiate the outer at u, then multiply by u'. For sin(3x²): outer'=cos, inner u=3x², u'=6x → f'=cos(3x²)·6x.