Problem 54 - Brief Applied Calculus

Question

Find an expression for the derivative of the composition of three functions, \(\frac{d}{d x} f(g(h(x)))\). [Hint: Use the Chain Rule twice.]

Step-by-Step Solution

Answer

The derivative is \(f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x)\).

Step 1: Identify the Outer Functions

To solve for the derivative of the composition of three functions, first identify the composition as three nested functions: the outer function is \(f\), the middle function is \(g\), and the innermost function is \(h\). Therefore, the overall function is written as \(f(g(h(x)))\).

Step 2: Apply the Chain Rule to the Composition

Apply the chain rule to differentiate \(f(g(h(x)))\). The chain rule states that the derivative of a composition \(f(u)\) with respect to \(x\) is \(f'(u) \cdot u'(x)\). Here, let \(u = g(h(x))\). Thus, the derivative with respect to \(x\) is \(f'(g(h(x))) \cdot \frac{d}{dx}(g(h(x)))\).

Step 3: Differentiate the Inner Function using the Chain Rule

Now, apply the chain rule again to differentiate \(g(h(x))\). Let \(v = h(x)\), thus \(g(h(x))\) becomes \(g(v)\). The derivative \(\frac{d}{dx}(g(h(x))) = g'(h(x)) \cdot h'(x)\).

Step 4: Combine the Results

Combine the results from Step 2 and Step 3 to get the derivative of the original composition. Substitute \(\frac{d}{dx}(g(h(x)))\) from Step 3 into the expression from Step 2:\[ \frac{d}{dx} f(g(h(x))) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \]

Step 5: Final Expression

The expression obtained in the previous step is the final expression for the derivative of the composition of the three functions. The derivative is given by:\[ \frac{d}{dx} f(g(h(x))) = f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \]

Key Concepts

Derivative of composite functions

Understanding the derivative of composite functions is an essential part of calculus. When dealing with functions that are composed of other functions, such as in the form \( f(g(h(x))) \), we must utilize the Chain Rule. This powerful tool helps us navigate through the layers of functions to find the overall rate of change. Initially, we identify that we are working with a cascade of functions, often referred to as 'nested functions.' In this example, \( f(g(h(x))) \) involves three functions: \( f \) is the outermost function, followed by \( g \), and \( h \) as the innermost function, applied to \( x \). To differentiate such a composition, the derivative needs to account for the change at each level of the function, passing through each layer one by one, using the Chain Rule at each step.

Differentiation techniques

Differentiation techniques are crucial tools in calculus for finding the rate of change of functions. The Chain Rule is one such technique that is indispensable when dealing with nested functions. - **Basics of the Chain Rule:** - For two functions, if you have \( y = f(u) \) and \( u = g(x) \), the derivative of \( y \) with respect to \( x \) is \( \frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx} \). - For three functions, like in our case, the Chain Rule extends further: we need to differentiate as follows: - First, identify \( u = g(h(x)) \) and use the Chain Rule for the outer function: \( f'(u) \cdot \frac{du}{dx} \). - Then differentiate \( g(h(x)) \) itself as a composition: use the Chain Rule again.By systematically applying the Chain Rule twice, we successfully differentiate composite functions such as \( f(g(h(x))) \), leading to the expression \( f'(g(h(x))) \cdot g'(h(x)) \cdot h'(x) \).

Nested functions in calculus

Nested functions in calculus refer to functions that are within other functions, such as \( f(g(h(x))) \). These types of functions require careful differentiation because each function must be considered in the order they are applied from innermost to outermost.When you see a function deeply nested like this, remember:- **Stepwise Approach:** - Begin with the innermost function, find its derivative, and move outward. - For \( f(g(h(x))) \), start by differentiating \( h(x) \) first. - Continue with \( g(h(x)) \), treating \( h(x) \) as a variable. - Finally, apply the derivative of \( f \), considering its input as \( g(h(x)) \).This approach ensures that you correctly account for the derivative at each level, thereby effectively handling the "nested layers" that are typical in calculus problems involving composite functions.