Problem 21 - Calculus - AP Edition

Question

Find \(f^{\prime}(x)\) $$f(x)=2 \sec ^{2}\left(x^{7}\right)$$

Step-by-Step Solution

Answer

Use the chain rule to differentiate the composite function, applying it to \( \sec^2(x^7) \).

Step 1: Understand the Function

The given function is \( f(x) = 2 \sec^2(x^7) \). Here, \( \sec^2(u) \) is a composite function where \( u = x^7 \). To differentiate \( \sec^2(u) \), we need to use the chain rule.

Step 2: Apply the Chain Rule

The chain rule states that if a function \( y = g(u) \) is a composite function of \( u = h(x) \), then \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). In this case, \( g(u) = \sec^2(u) \) and \( h(x) = x^7 \).

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate composite functions. When you have a function nested inside another, like in our example, it's critical to apply the chain rule to find the derivative correctly.

To understand this better, imagine you're peeling an onion. Each layer must be peeled back in succession to reach the core. Similarly, the chain rule allows us to "peel" each function layer at a time. The chain rule formula is:

• Let \( y = g(u) \) and \( u = h(x) \).
• Then, the derivative \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).

This means you first find the derivative of the outer function \( g(u) \) with respect to the inner function \( u \), and then multiply by the derivative of the inner function \( u = h(x) \) with respect to \( x \).

In our exercise, \( g(u) = \sec^2(u) \) and \( u = x^7 \), making the chain rule an appropriate choice for differentiation.

Composite Functions

Composite functions are like a "function within a function." You can think of them as processes where the result of one function becomes the input for another. In mathematical terms, a composite function \( f(x) = g(h(x)) \) involves a function \( g \) of another function \( h \).

In the context of the given problem, we have the composite function \( f(x) = 2 \sec^2(x^7) \). Here, \( \sec^2 \) is applied to \( x^7 \), making \( x^7 \) the inside function and \( \sec^2 \) the outside function. Understanding this structure is crucial when applying the chain rule and dealing effectively with each layer of the composition.

• In our function, \( f(x) = 2 \sec^2(x^7) \), \( x^7 \) acts as the input to the \( \sec^2 \) function.
• To differentiate, first consider the derivative of \( \sec^2(u) \) and then account for \( u = x^7 \).

Having a firm grasp of composite functions makes differentiating them, especially using the chain rule, much more manageable.

Trigonometric Functions

Trigonometric functions are a class of functions that relate angles to ratios of side lengths in right triangles. Common ones include sine, cosine, and tangent, but others like secant, cosecant, and cotangent are also critical in calculus.

In the exercise here, \( \sec^2(x) \) is the trigonometric function involved. The secant function, \( \sec(x) \), is the reciprocal of \( \cos(x) \), defined by \( \sec(x) = \frac{1}{\cos(x)} \). Consequently, \( \sec^2(x) = \left(\frac{1}{\cos(x)}\right)^2 \).

• The derivative of \( \sec(x) \) with respect to \( x \) is \( \sec(x)\tan(x) \), a result you will frequently use when applying calculus to trigonometric functions.
• For \( \sec^2(x) \), the derivative can be found by first using the power rule and then applying the chain rule.

Thus, recognizing and differentiating trigonometric functions is essential in solving our given problem.