Problem 13 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

In Problems 13 through 18, find \(h^{\prime}(x) .\) Assume that \(f\) and \(g\) are differentiable on \((-\infty, \infty)\). Your answers may be in terms of \(f, g, f^{\prime}\), and \(g^{\prime}\). $$ h(x)=f(\ln x)-\ln (f(x)) $$

Step-by-Step Solution

Answer

The derivative of the function \(h(x)=f(\ln x)-\ln (f(x))\) is \(h'(x)=f'(\ln x) \cdot \frac{1}{x} - \frac{f'(x)}{f(x)}\).

Step 1: Separate the Function

The function \(h(x)\) can be split up into two different terms for easier differentiation: \(h(x) = f(\ln x)\) and \(- \ln(f(x))\). These will be tackled separately.

Step 2: Differentiate the First Term

Differentiate the first term using the Chain Rule. The Chain Rule states that the derivative of a composition of functions is the derivative of the outer function multiplied by the derivative of the inner function. Therefore, the derivative of \(f(\ln x)\) is \(f'(\ln x) \cdot \frac{1}{x}\).

Step 3: Differentiate the Second Term

Now we differentiate the second term \(-\ln(f(x))\). Again, we will use the Chain Rule. The derivative of \(-\ln(f(x))\) is \(-\frac{1}{f(x)} \cdot f'(x)\).

Step 4: Combine the Derivatives

Now we can put the two derivatives together to find the derivative of the original function \(h(x)\). Therefore, the derivative of \(h(x) = f'(\ln x) \cdot \frac{1}{x} - \frac{f'(x)}{f(x)}\).

Key Concepts

Chain Rule

The Chain Rule is an essential tool in calculus for differentiating composite functions. It allows us to find the derivative of a function based on the derivatives of its inner and outer functions. When you have a composition, like \( f(g(x)) \), the Chain Rule provides a way to take into account how both \( f \) and \( g \) change.
Here's a simple approach to understand the Chain Rule:

• Differentiate the outer function. Leave the inside function untouched.
• Multiply this result by the derivative of the inside function.

In the exercise example, for the first term \( f(\ln x) \), the outer derivative is \( f'(\ln x) \) and you multiply this by the derivative of \( \ln x \), which is \( \frac{1}{x} \). This gives \( f'(\ln x) \cdot \frac{1}{x} \). For the term \( -\ln(f(x)) \), the chain rule also plays a vital role, with \( -\frac{1}{f(x)} \) as the outer function's derivative, and \( f'(x) \) as the derivative of the inner function. This results in \(-\frac{f'(x)}{f(x)}\).
The Chain Rule is necessary whenever you differentiate composite functions. By breaking down the function into parts, you can apply the rule successfully.

Composite Functions

Composite functions involve combining two functions to make a new one. If you have two functions \( f(x) \) and \( g(x) \), you can compose them to form \( f(g(x)) \). The inner function \( g(x) \) becomes the input for the outer function \( f \). In other words, you are applying \( f \) to the result of \( g \). For instance, if \( f(x) = x^2 \) and \( g(x) = \ln x \), the composite function \( f(g(x)) \) becomes \( (\ln x)^2 \). Calculus requires special rules to handle derivatives of these composite functions. The Chain Rule is what helps us differentiate this kind of function by dealing separately with the outer and inner functions.
To tackle derivatives of composite functions like in our problem,

• Identify the outer and inner functions. This helps clarify the path to apply the Chain Rule.
• Understand each component individually before combining them.

By recognizing composite functions and using the Chain Rule, you can efficiently find derivatives of complex expressions.

Derivatives of Logarithmic Functions

Logarithmic functions have unique derivative rules. They are treated distinctively in calculus due to their special properties. The most basic logarithmic function is \( \ln x \) (the natural logarithm). Its simple derivative is \( \frac{1}{x} \). However, complications arise when the argument of the logarithm itself is a function, such as in \( \ln(f(x)) \).
Here's how to differentiate logarithmic functions using the Chain Rule:

• Start with the standard rule for the natural logarithm, \( \frac{1}{x} \) for \( \ln x \).
• If you have \( \ln(f(x)) \), use the Chain Rule to multiply \( \frac{1}{f(x)} \) by \( f'(x) \). This results in \( \frac{f'(x)}{f(x)} \).

In our solved exercise, the term \( \ln(f(x)) \) requires this method to differentiate it, leading to \( -\frac{f'(x)}{f(x)} \). Similarly, differentiating terms with natural logarithms calls for attention to both the function inside the logarithm and its derivative. Understanding logarithmic differentiation helps deal with complex expressions efficiently. It is crucial whenever you encounter logs in calculus problems.