Derivatives of Trigonometric Functions: Sine and Cosine

The Two Rules You Need to Know

The derivatives of sine and cosine are among the most important rules in AP Calculus. They appear constantly in differentiation problems, especially when trigonometric functions are combined with the chain rule, product rule, or quotient rule.

The two basic rules are:

Notice the important difference: sine differentiates to positive cosine, while cosine differentiates to negative sine.

Why These Rules Work

The derivative of a function tells us its instantaneous rate of change. For the sine and cosine functions, their rates of change are themselves trigonometric functions.

As the angle changes, the slope of the sine curve follows the cosine curve. Similarly, the slope of the cosine curve follows the negative of the sine curve.

This relationship is especially useful because it means that differentiating a trigonometric function does not make the function fundamentally more complicated. Instead, sine and cosine cycle into one another under differentiation.

The Most Important Pattern

When the argument of sine or cosine is simply , the derivative rules can be applied directly. But if the argument contains something more complicated, such as or , you need the chain rule.

For a composite function, the general rules become:

In other words, differentiate the outside trigonometric function first, then multiply by the derivative of whatever is inside it.

A Step-by-Step Method

Identify the trigonometric function. Is the outside function sine or cosine?
Differentiate the outside function. Sine becomes cosine; cosine becomes negative sine.
Keep the inside expression unchanged. Replace the argument of sine or cosine without differentiating it yet.
Multiply by the derivative of the inside expression. This is the chain rule.
Simplify if appropriate.

Worked Examples

Example 1: Derivative of Sine

Differentiate .

Step 1: The function is sine, and its argument is simply .

Step 2: Use the basic sine derivative rule: .

Result:

Example 2: Derivative of Cosine

Differentiate .

Step 1: The function is cosine, with argument .

Step 2: Cosine differentiates to negative sine: .

Result:

Example 3: Sine with the Chain Rule

Differentiate .

Step 1: Outside function: . Inside function: .

Step 2: Differentiate the outside: .

Step 3: Differentiate the inside: .

Step 4: Multiply: .

Result:

Example 4: Cosine with the Chain Rule

Differentiate .

Step 1: Outside function: . Inside function: .

Step 2: Cosine differentiates to negative sine, giving .

Step 3: The derivative of the inside is .

Step 4: Multiply: .

Result:

Example 5: Trigonometric Function with a Polynomial Inside

Differentiate .

Step 1: Outside function: . Inside function: .

Step 2: Differentiate the outside while leaving the inside unchanged: .

Step 3: Differentiate the inside: .

Step 4: Apply the chain rule: .

Result:

Why the Negative Sign Matters for Cosine

One of the most common AP Calculus mistakes is forgetting the negative sign when differentiating cosine.

Remember the cycle:

So:

Sine becomes cosine.
Cosine becomes negative sine.

A useful memory device is simply: sin → cos, cos → −sin.

Common Mistakes

Forgetting the negative sign

The derivative of cosine is , not . That negative sign is part of the derivative rule and cannot be dropped.

Forgetting the chain rule

For , the derivative is not just . You must also multiply by the derivative of the inside function.

Changing the inside function

When differentiating , the inside remains when you differentiate the outside. The chain rule changes the multiplier, not the original argument.

Confusing trigonometric functions with inverse trigonometric functions

The derivative rules for and are different from the rules for inverse functions such as and . Make sure you identify the function correctly before differentiating.

Practice Problems

Question 1

Differentiate .

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Question 2

Differentiate .

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Question 3

Differentiate .

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Question 4 (Combined Rules)

Differentiate .

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