Differentiating Inverse Trigonometric Compositions

What Inverse Trigonometric Compositions Are

The inverse trigonometric functions , , and each have their own derivative formula. But on AP Calculus problems, the input to the inverse trigonometric function is often more complicated than simply .

For example, is a composition: the outside function is , while the inside function is . That means the chain rule must be applied.

The same idea works for expressions such as and . The inverse-trig derivative formula gives you the derivative of the outside function; the chain rule then requires you to multiply by the derivative of the inside function.

The Three Derivative Formulas

These are the three inverse-trigonometric derivatives you should know for AP Calculus:

A useful way to remember the pattern is that the derivative of an inverse-trig function produces a square root or a quadratic expression in the denominator. The exact form depends on which inverse-trig function you are differentiating.

The Chain Rule Version

If the input is a function rather than simply , replace the in the basic formula with the entire inside function, then multiply by its derivative.

In other words, the process is: use the inverse-trig derivative formula, keep the inside function intact, then multiply by the derivative of the inside.

A Step-by-Step Method

Identify the inverse-trig function. Is the outside function arcsin, arccos, or arctan?
Identify the inside function. Treat everything inside the inverse-trig function as one expression.
Write the appropriate inverse-trig derivative formula. Do not forget the square root or quadratic expression in the denominator.
Multiply by the derivative of the inside function. This is the chain-rule step.
Simplify. Factor, combine constants, or rewrite the result when appropriate.

Worked Examples

Example 1: Arcsin Composition

Differentiate .

Step 1: The outside function is , and the inside function is .

Step 2: Use the arcsin derivative formula:

Step 3: Differentiate the inside:

Step 4: Multiply by the inside derivative:

Result:

Example 2: Arccos Composition

Differentiate .

Step 1: Outside function: . Inside function: .

Step 2: The derivative of arccos is negative:

Step 3: Differentiate the inside:

Step 4: Apply the chain rule:

Result:

Example 3: Arctan Composition

Differentiate .

Step 1: Outside function: . Inside function: .

Step 2: Apply the arctan derivative formula:

Step 3: Differentiate the inside:

Step 4: Multiply by the inside derivative:

Result:

Example 4: Inverse Trig with a Polynomial Inside

Differentiate .

Step 1: The outside function is , and the inside function is .

Step 2: Apply the arcsin formula:

Step 3: Differentiate the inside:

Step 4: Multiply:

Result:

Example 5: Product Rule + Inverse-Trig Chain Rule

Differentiate . This example combines the product rule with an inverse-trigonometric composition, which is the kind of layered differentiation that appears frequently in AP Calculus.

Step 1: Use the product rule:

Step 2: The derivative of is .

Step 3: For , use the chain rule:

Step 4: Substitute both derivatives into the product rule:

Result:

Recognizing the Chain Rule Quickly

One of the easiest ways to avoid mistakes is to look at the argument of the inverse-trig function before doing any algebra.

Here the inside function is simply , whose derivative is 1. So the chain-rule multiplier does not visibly change the answer.

But compare that with . The inside function is now , whose derivative is . That factor must appear in the final derivative.

Common Mistakes

Forgetting the chain-rule factor

The most common error is to memorize and then use it unchanged for . The correct derivative must also include the derivative of the inside function.

Forgetting the negative sign for arccos

The derivative of arccos is negative: . It is easy to remember the arcsin formula and accidentally use the same sign for arccos.

Confusing inverse trig with reciprocal trig

means inverse sine, not reciprocal sine. The reciprocal of sine is cosecant, written . These are completely different functions.

Putting the inside function in the wrong place

For arcsin, the denominator is , where the entire inside function replaces . For arctan, the denominator is . Keep the entire composition intact before simplifying.

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

Differentiate .

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Question 5 (Combined Rules)

Differentiate .

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