Problem 109 - Calculus and its applications

Question

Differentiate. $$ g(x)=[\ln (x+5)]^{4} $$

Step-by-Step Solution

Answer

The derivative is \( g'(x) = \frac{4[\ln(x+5)]^3}{x+5} \).

Step 1: Identify the Function

The function given is \( g(x) = [\ln(x+5)]^4 \). This is a composition of two functions: the outer function \( u(t) = t^4 \) and the inner function \( v(x) = \ln(x+5) \).

Step 2: Apply the Chain Rule

To differentiate a composite function, we use the chain rule, which states \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \). Here, \( y = [\ln(x+5)]^4 \). We need to find \( \frac{dy}{du} \) and \( \frac{du}{dx} \).

Step 3: Differentiate the Outer Function

Differentiate the outer function \( u(t) = t^4 \) with respect to \( t \). This gives \( \frac{du}{dt} = 4t^3 \).

Step 4: Differentiate the Inner Function

Differentiate the inner function \( v(x) = \ln(x+5) \) with respect to \( x \). This gives \( \frac{du}{dx} = \frac{1}{x+5} \).

Step 5: Combine Results Using the Chain Rule

Using the chain rule, substitute \( u = \ln(x+5) \) into the derivative of the outer function: \( 4[\ln(x+5)]^3 \cdot \frac{1}{x+5} \).

Step 6: Write the Final Derivative

Combine the terms to get the derivative of \( g(x) \): \[ g'(x) = \frac{4[\ln(x+5)]^3}{x+5} \]

Key Concepts

Composite Function

When dealing with composite functions, we are essentially combining two or more functions to create a new one. In the exercise above, we have the function \( g(x) = [\ln(x+5)]^4 \), which is a composition of two simpler functions: \( u(t) = t^4 \) (the outer function) and \( v(x) = \ln(x+5) \) (the inner function). This is an essential concept for differentiation because it helps us understand how to take derivatives of more complex functions by breaking them down.

• The outer function \( u(t) = t^4 \) applies an operation to the output of the inner function.
• The inner function \( v(x) = \ln(x+5) \) is the first function applied to the input variable \( x \).

By understanding the structure of composite functions, we can apply the chain rule efficiently to differentiate them.

Logarithmic Differentiation

Logarithmic differentiation is a technique especially useful when dealing with products, powers, or composites of functions involving logarithms. In our problem of differentiating \( g(x) = [\ln(x+5)]^4 \), we encounter a logarithmic function as part of the composite.
When you have a function where a logarithm is raised to a power like \( [\ln(x+5)]^4 \), it requires careful application of the derivative. Notice that each component can be tackled by applying specific rules of differentiation:

• For \( [f(x)]^n \), use the power rule which yields \( n[f(x)]^{n-1} \).
• For \( \ln(u) \), utilize the derivative \( \frac{1}{u} \cdot u' \).

These principles make logarithmic differentiation a powerful method, allowing us to simplify complex expressions into more manageable derivatives.

Calculus Problem-solving

Solving calculus problems like differentiating a function involves a systematic approach. It requires identifying the structure of the function and applying appropriate differentiation techniques. In our exercise, we used the steps of the chain rule to solve the problem efficiently.

• Firstly, distinguish between the inner and outer functions. Recognizing these helped simplify our approach.
• Use the chain rule: If you have a composite function \( f(g(x)) \), differentiate \( f \) first with respect to the inner function and then multiply by the derivative of the inner function.
• Calculate step by step. Break down each component of the function, differentiate separately, and then combine.

This deliberate problem-solving approach is key to handling any calculus problems effectively. Step-by-step problem-solving not only prevents errors but also enhances understanding, leading to improved skills in calculus troubleshooting.