Problem 48 - Calculus - AP Edition

Question

Find \(d y / d x\). $$y=\sec ^{-1}\left(x^{5}\right)$$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{5}{x\sqrt{x^{10} - 1}} \) for \( x > 0 \).

Step 1: Understand the problem

We need to find the derivative of the function \( y = \sec^{-1}(x^5) \) with respect to \( x \). This means we want to compute \( \frac{dy}{dx} \).

Step 2: Use the Chain Rule

Observe that \( y = \sec^{-1}(u) \), where \( u = x^5 \). To differentiate \( y \) with respect to \( x \), we first need \( \frac{dy}{du} \) and \( \frac{du}{dx} \). The chain rule gives us \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

Step 3: Differentiate the Inverse Secant

The derivative of \( \sec^{-1}(u) \) with respect to \( u \) is \( \frac{1}{|u|\sqrt{u^2 - 1}} \). Therefore, \( \frac{dy}{du} = \frac{1}{|x^5|\sqrt{(x^5)^2 - 1}} \).

Step 4: Differentiate the Power Function

Find \( \frac{du}{dx} \), where \( u = x^5 \). The derivative is \( \frac{du}{dx} = 5x^4 \).

Step 5: Apply the Chain Rule

Combine the derivatives using the chain rule: \( \frac{dy}{dx} = \frac{1}{|x^5|\sqrt{(x^5)^2 - 1}} \times 5x^4 \).

Step 6: Simplify the Expression

Simplify the expression: \( \frac{dy}{dx} = \frac{5x^4}{|x^5|\sqrt{x^{10} - 1}} \). Observing that \( |x^5| = x^5 \) if \( x > 0 \) and \( |x^5| = -x^5 \) if \( x < 0 \), the expression further simplifies for \( x > 0 \) to \( \frac{5}{x\sqrt{x^{10} - 1}} \).

Key Concepts

Chain Rule

The Chain Rule is an essential tool in calculus for finding the derivative of composite functions. A composite function is one in which one function is nestled inside another, such as in our exercise where we have \( y = \sec^{-1}(x^5) \). Here, the outer function is the inverse secant, and the inner function is the power function \( x^5 \).
To apply the Chain Rule, you first differentiate the outer function concerning its inner function and then multiply it by the derivative of the inner function. The formula is:

• \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \)

In this context, we first found \( \frac{dy}{du} \) by differentiating the inverse secant function and then \( \frac{du}{dx} \) by differentiating \( x^5 \). Finally, multiplying these derivatives gives us the full derivative of the composite function.

Inverse Trigonometric Functions

Inverse Trigonometric Functions, like \( \sec^{-1}(x) \), reverse the roles of input and output compared to their standard trigonometric counterpart. When it comes to differentiation, these functions have specific rules you must apply. In our exercise, we needed the derivative of \( \sec^{-1}(u) \) with respect to \( u \).
The derivative is:

• \( \frac{dy}{du} = \frac{1}{|u|\sqrt{u^2 - 1}} \)

This formula arises because of the nature of the secant function, which involves ratios and angles that create these radical expressions. When you solve these derivatives in practical problems, be cautious about domain restrictions and absolute value considerations which ensure the range of real numbers stays defined.
This definitive method helps simplify finding derivatives of these often complicated-looking functions.

Power Rule

The Power Rule is one of the simplest yet powerful rules of differentiation. It allows us to find the derivative of any function that can be expressed as a power of \( x \). For a function \( u = x^n \), the Power Rule states that:

• \( \frac{du}{dx} = nx^{n-1} \)

In the exercise, we applied the Power Rule to find the derivative of the function \( x^5 \). By replacing \( n \) with 5 in the formula, we found:

• \( \frac{du}{dx} = 5x^4 \)

Understanding the Power Rule is crucial for handling functions involving exponents, making it a fundamental component in calculus. Its simplicity makes it easy to apply, and when used within the Chain Rule for composite functions, it unlocks the door to solving more complex derivative problems.