91Ó°ÊÓ

When differentiating functions that are composed of other functions, the chain rule is a critical tool in calculus. The chain rule is used to differentiate composite functions. If you have a function, say \( y = f(g(x)) \), the chain rule tells you how to find the derivative of \( y \) with respect to \( x \).

The process involves taking the derivative of the outer function \( f \) and multiplying it by the derivative of the inner function \( g \). This can be expressed as:

• Outer derivative: \( f'(g(x)) \)
• Inner derivative: \( g'(x) \)
• Combined: \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \)

For example, in finding the first derivative \( \frac{dy}{dx} \) of the function \( y = e^{(e^x)} \), the chain rule is applied by differentiating the outer function \( e^u \) and multiplying it by the derivative of the inner function \( u = e^x \), leading to the expression \( \frac{dy}{dx} = e^{(e^x)} \cdot e^x \).

The product rule is utilized when you need to differentiate the product of two or more functions.

Suppose you have functions \( u(x) \) and \( v(x) \), and you wish to differentiate their product, which is expressed as \( y = u(x) \cdot v(x) \). The product rule is formulated as:

• \( u'(x) \cdot v(x) + u(x) \cdot v'(x) \)

In the context of finding the second derivative \( \frac{d^2y}{dx^2} \) for our function \( y = e^{(e^x)} \cdot e^x \), apply the product rule. Let \( u = e^{(e^x)} \) and \( v = e^x \).
Differentiating \( u \) involves using the chain rule, just like when finding \( \frac{dy}{dx} \). The product rule then combines the derivatives of "\( u \) multiplied by \( v \)" and "\( u \) itself." Therefore, you end up with the expression \:\[\frac{d^2y}{dx^2} = (e^{(e^x)} \cdot e^x) \cdot e^x + e^{(e^x)} \cdot e^x.\]

A composite function occurs when one function is nested inside another. Understanding how composite functions differ from simple functions is essential for applying calculus techniques like the chain rule efficiently.

An example of a composite function is \( y = e^{(e^x)} \). Here, the inner function is \( e^x \) and the outer function is the exponential \( e^u \).

• The inner function \( g(x) = e^x \).
• The outer function \( f(u) = e^u \), with \( u = g(x) \).

Composite functions require careful analysis because the derivative of the entire function \( y \) relies on derivatives of both the inner \( g(x) \) and outer functions \( f(u) \). To differentiate the composite function, you apply concepts such as the chain rule, to systematically break down each part of the composition, ensuring accurate results for both the first and second derivatives. In our specific example, understanding it as a composite function helps logically apply the chain and product rules to reach the final expression for the second derivative.