The Chain Rule & Trig Derivatives

You’ll be able to

Differentiate composite functions with the chain rule
State and apply the derivatives of the six trigonometric functions
Combine the chain rule with trig, power, and product rules

The chain rule for composites

A composite function f(g(x)) is a function inside another function. The chain rule differentiates it: d/dx[f(g(x))] = f′(g(x))·g′(x) — the derivative of the outer function (evaluated at the inside, left alone) times the derivative of the inside. The factor g′(x) is the piece students forget; it is often called the "inside derivative". Peel the layers from the outside in, and multiply each layer’s derivative as you go.

Derivatives of the trig functions

The six trig derivatives must be memorized. The two anchors are d/dx[sin x] = cos x and d/dx[cos x] = −sin x — note the minus sign on cosine. From the quotient rule follow d/dx[tan x] = sec²x and d/dx[cot x] = −csc²x, plus d/dx[sec x] = sec x·tan x and d/dx[csc x] = −csc x·cot x. A memory aid: every "co-" function (cos, cot, csc) carries a negative sign.

Chain rule meets trig

Almost every trig derivative in practice is a composite, because the angle is rarely just x. For sin(3x²) the outside is sine and the inside is 3x², so the derivative is cos(3x²)·(6x). Always multiply by the derivative of the angle. This is the single most common place to lose points — writing cos(3x²) and forgetting the ·6x factor.

Chain rule

d/dx[f(g(x))] = f′(g(x)) · g′(x)

Outer derivative (inside untouched) times inner derivative. In Leibniz form, dy/dx = dy/du · du/dx.

Core trig derivatives

d/dx[sin x] = cos x · d/dx[cos x] = −sin x · d/dx[tan x] = sec²x

Also: d/dx[sec x] = sec x·tan x, d/dx[csc x] = −csc x·cot x, d/dx[cot x] = −csc²x.

Worked example

Differentiate y = sin(x³ + 1).

1.Identify the composition: the outer function is sine, the inner function is u = x³ + 1.
2.Derivative of the outer, leaving the inside alone: cos(x³ + 1).
3.Derivative of the inner: d/dx[x³ + 1] = 3x².
4.Multiply the two by the chain rule.

Answer: y′ = cos(x³ + 1)·3x² = 3x²·cos(x³ + 1).

Watch out

The most common chain-rule error is dropping the inside derivative. d/dx[sin(5x)] is 5cos(5x), not cos(5x). If the angle (or the inside) is anything other than a bare x, that extra factor is required.

Checkpoint

What is d/dx[cos(4x)]?

Checkpoint

If y = (x² + 1)⁵, what is dy/dx?

On the exam

Think of the chain rule as peeling an onion: differentiate the outermost layer first, keep the inside intact, then multiply by the derivative of the inside, repeating for nested compositions. Track every layer and you will never drop a factor.

Answer the 2 checkpoints as you read.