How to Differentiate Trigonometric Functions

Memorize the six base derivatives

For angles measured in radians,

$$\begin{aligned} \frac{d}{dx}(\sin x)&=\cos x, &\frac{d}{dx}(\cos x)&=-\sin x,\\ \frac{d}{dx}(\tan x)&=\sec^2x, &\frac{d}{dx}(\cot x)&=-\csc^2x,\\ \frac{d}{dx}(\sec x)&=\sec x\tan x, &\frac{d}{dx}(\csc x)&=-\csc x\cot x. \end{aligned}$$

The negative signs belong to the cosine, cotangent, and cosecant derivatives. Keeping those three together is a useful error check.

Combine the base rules with ordinary derivative rules

A trigonometric term may sit inside a sum, product, quotient, or composition. First identify the outer algebraic structure, then apply the matching rule.

For a composition, the chain rule adds the derivative of the inner function. For example,

$$\frac{d}{dx}\sin(4x^2) =\cos(4x^2)\cdot8x.$$

For products and quotients, keep the trigonometric derivative attached to its original factor before simplifying.

Example with a different function

Let

$$g(x)=x^2\sec x.$$

The product rule gives

$$\begin{aligned} g'(x) &=(2x)\sec x+x^2(\sec x\tan x)\\ &=\boxed{2x\sec x+x^2\sec x\tan x}. \end{aligned}$$

Check the original domain

A derivative formula is valid only where the original function is defined. Expressions containing $\tan x$ or $\sec x$ exclude points where $\cos x=0$; expressions containing $\cot x$ or $\csc x$ exclude points where $\sin x=0$. Algebraic simplification should not silently restore excluded points.