Calculus I · Unit 2A · lesson

Derivatives of the Other Trigonometric Functions

Concept

Learning objectives

Differentiate all six trigonometric functions and connect the rules to quotient and product identities.

Tangent, Cotangent, Secant, and Cosecant

Explanation

Before the formulas

In Derivatives of the Other Trigonometric Functions, begin by asking what the original function does. Where does it rise, fall, flatten, or grow in proportion to itself? The derivative formula should match that qualitative behavior. A missing minus sign in the cosine derivative, for example, contradicts the fact that cosine decreases immediately to the right of zero.

When several special functions appear together, separate recognition from computation. Identify each basic derivative, note any composition requiring the chain rule, and then combine the pieces. A short verbal plan keeps a crowded formula from becoming a guessing contest.

Explanation

Build the remaining trig derivatives from sine and cosine

Tangent, cotangent, secant, and cosecant do not require four unrelated derivations. Rewrite them using sine and cosine, then use quotient or product rules. The resulting formulas inherit both the geometry of the unit circle and the algebra of reciprocals.

Domain restrictions matter. A formula involving $\sec x$ is meaningful only where $\cos x\neq0$ , and the derivative cannot repair a point where the original function is undefined.

The remaining trigonometric derivatives do not need six unrelated proofs. Tangent and cotangent are quotients; secant and cosecant are reciprocals. The product and quotient rules, together with the Pythagorean identities, generate the entire table.

Rebuilding a forgotten formula from identities is slower than perfect memory but much faster than inventing a sign. It also explains where the domain restrictions come from: a trigonometric derivative cannot be used where the original function is undefined.

Using $\tan x=\sin x/\cos x$ and the quotient rule gives

\frac{d}{dx}\tan x=\sec^2x.

The complete table is

\begin{array}{c|c} \text{function}&\text{derivative}\\ \sin x&\cos x\\ \cos x&-\sin x\\ \tan x&\sec^2x\\ \cot x&-\csc^2x\\ \sec x&\sec x\tan x\\ \csc x&-\csc x\cot x \end{array}

Method

A sign pattern worth understanding

The cofunctions beginning with the letter "c" are not all negative. The negative rules are cosine, cotangent, and cosecant. Secant is positive. It is better to rebuild a forgotten rule from identities than to rely on a panicked chant.

Guided walkthrough

Differentiate a quotient containing tangent

Differentiate

y=\frac{\tan x}{x^2}.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

Differentiate a secant product

\frac{d}{dx}[x\sec x]=\sec x+x\sec x\tan x.

Both terms are required by the product rule.

Exercise

Differentiate $5\tan x-2\cot x$ .

Exercise

Differentiate $x^2\csc x$ .

Exercise

Differentiate $\sec x/(x+1)$ .

Exercise

Find the tangent line to $y=\tan x$ at $x=\pi/4$ .

After the explanation

Use the section idea

Reading lens

Connect each derivative formula to the function's characteristic growth, periodicity, or inverse relationship.

Mental model

Special-function rules preserve recognizable shapes while scaling them by a function-specific factor.

Decision

Identify the function family first, then check whether a composition requires the chain rule too.

Common trap

Using a power rule on an exponential or forgetting base and domain conditions for logarithms.

Check yourself

Can you distinguish a power, exponential, logarithmic, and trigonometric derivative at a glance?

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