Calculus I · Unit 2A · lesson

Chain Rule with Trigonometric Functions

Concept

Learning objectives

Differentiate trigonometric functions with nontrivial inputs.

Trigonometric Compositions

Explanation

Before the formulas

Composition is the hidden architecture of Chain Rule with Trigonometric Functions. An expression may look like one formula while actually containing several function machines. Draw or list the stages before differentiating. This is especially important when product, quotient, and chain rules appear together, because the outer algebraic connection and the inner compositions must both be respected.

Do not simplify away useful structure too early. A factored derivative often shows each chain factor more clearly and is easier to verify. Simplify after the calculus unless rewriting first genuinely reduces the number of rules required.

Explanation

Trig functions often hide a changing angle

The derivative of $\sin x$ is $\cos x$ , but $\sin(5x^2)$ is not receiving the input $x$ directly. Its angle changes at rate $10x$ , so the final derivative must include that factor.

Read "sine of something" as an outer sine function attached to an inner angle function. This language makes the chain rule visible before any symbols are manipulated.

A trigonometric formula such as $\sin(4t)$ describes an oscillation whose phase changes four times as fast as $t$ . The chain-rule factor $4$ records that increased frequency. Without it, the derivative would describe the wrong physical speed.

The same idea applies to shifted and nonlinear phases. In $\cos(t^2)$ , the angle itself accelerates because $t^2$ changes at rate $2t$ . The derivative combines the local slope of cosine with that changing phase rate.

The basic patterns are

\frac{d}{dx}\sin u=\cos u\,u', \qquad \frac{d}{dx}\cos u=-\sin u\,u',

\frac{d}{dx}\tan u=\sec^2u\,u',

with analogous rules for the remaining trig functions.

Guided walkthrough

A sine of a quadratic

Differentiate

y=\sin(x^2+3x).

Answer reveal

Worked solution

Your setup or answer

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Worked example

A squared trig function

For $y=\sin^2x=(\sin x)^2$ ,

y'=2\sin x\cos x=\sin(2x).

The expression $\sin^2x$ is a composition: square the value of sine.

Worked example

Tangent with a reciprocal input

y=\tan(1/x),

y'=\sec^2(1/x)\left(-\frac1{x^2}\right) =\boxed{-\frac{\sec^2(1/x)}{x^2}}.

Interactive checkchain-trig-01

Differentiate $\sin(3x^2)$ .

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Show hint

Differentiate sine first, then multiply by the derivative of 3x^2.

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Application

A pendulum's angular speed

For a small oscillation, suppose

\theta(t)=0.20\cos(3t+0.4)

radians. Then

\theta'(t)=-0.60\sin(3t+0.4)

radians per second. The factor $3$ is the phase rate. Omitting it would understate every angular speed by a factor of three.

After the explanation

Use the section idea

Reading lens

Read nested functions from the outside inward, but multiply local response factors through every layer.

Mental model

A small input change passes through a sequence of machines; the total response multiplies the response at each stage.

Decision

List the layers, differentiate one layer at a time, and stop only when every input-dependent layer contributes.

Common trap

Differentiating the outside and leaving the inside unchanged without its derivative factor.

Check yourself

Can you annotate every factor in your derivative with the layer that produced it?

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