91Ó°ÊÓ

The chain rule is a fundamental concept in calculus that is integral to understanding how to differentiate complex functions. When we are dealing with a composition of functions, like in the case of finding the antiderivative of \( f(x) = \sec^2(-4x) \), the chain rule helps us determine how the rate of change of one function affects another within the composition. In simpler terms, if you have a function inside another function, the chain rule allows you to differentiate the outer function while multiplying it by the derivative of the inner function. For example:

• Consider a function \( y = \sec^2(u) \) where \( u = -4x \).
• According to the chain rule, if we are differentiating this, we first take the derivative of \( \sec^2(u) \) which is \( 2\sec(u)\tan(u) \), and then multiply it by the derivative of \( u \), which is \( -4 \).

However, finding an antiderivative requires reversing this process, often through a method called substitution, which is essentially the chain rule in reverse.

The substitution method is a powerful tool often used alongside the chain rule when working with integrals. This method typically simplifies the integration process by transforming a complex integral into a much simpler one. The substitution method involves setting a part of the integral equal to a new variable, often denoted as \( u \). This is particularly helpful when dealing with composite functions, like \( \sec^2(-4x) \), as it allows us to simplify the expression:

• First, identify the inner function. In our problem, this is \( -4x \).
• Set \( u = -4x \) and find \( dx \) in terms of \( du \). We do this by differentiating \( u \) to get \( du = -4 \, dx \), or \( dx = \frac{du}{-4} \).
• Substitute \( u \) and \( dx \) back into the integral to simplify it: \( \int \sec^2(u) \cdot -\frac{1}{4} \, du \).
• After substitution, it becomes a simple integral to solve: \( -\frac{1}{4} \int \sec^2(u) \, du \).
• The antiderivative of \( \sec^2(u) \) is \( \tan(u) \), which gives us \( -\frac{1}{4}\tan(u) + C \).

Finally, replace \( u \) with \( -4x \) to obtain \( \frac{1}{4} \tan(4x) + C \), ensuring we finish with the expression in terms of the original variable \( x \). This technique allows us to find the antiderivative more easily by dealing with simpler functions first.

Differential calculus deals with the concept of a derivative, providing a way to analyze changing quantities. When we talk about finding the antiderivative, which is essentially the reverse of taking a derivative, we engage with the concepts of differential calculus to understand how functions change. In the given problem, we started by recognizing the derivative relationship \( \frac{d}{dx} \tan(x) = \sec^2(x) \), allowing us to identify that the antiderivative of \( \sec^2(x) \) is \( \tan(x) \). This is a fundamental idea in calculus that links antiderivatives with the process of integration, the inverse of differentiation. Some key points about differential calculus include:

• It provides rules and techniques to find derivatives, such as the product rule, quotient rule, and, importantly for this exercise, the chain rule.
• It helps understand the rate at which quantities change, which is useful in various scientific and engineering contexts.
• While the derivative gives us insights into slopes and rates, the antiderivative helps us recapture the original function, which is often the objective in problems involving integrals.

Understanding differential calculus not only helps in solving derivative problems but also opens up methods for reversing these processes through integration, key to solving antiderivative issues like the one presented.