Problem 108 - Calculus and its applications

Question

Differentiate. $$ f(x)=\ln \left(x^{3}+1\right)^{5} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \ln((x^3+1)^5) \) is \( f'(x) = \frac{15x^2}{x^3 + 1} \).

Step 1: Recognize the Function Form

The given function is a composite function of the form \( f(x) = ext{something}^5 \). In this case, the expression inside the logarithm is \( x^3 + 1 \) raised to the 5th power. We will need to use the chain rule for logarithmic differentiation.

Step 2: Apply the Logarithmic Differentiation Formula

To differentiate \( f(x) = \ln((x^3 + 1)^5) \), use the property of logarithms: \( \ln(a^b) = b\ln(a) \). This simplifies the function to \( f(x) = 5\ln(x^3 + 1) \).

Step 3: Differentiate Using the Chain Rule

Differentiate \( f(x) = 5\ln(x^3 + 1) \) using the chain rule. Take the derivative of the outer function and multiply by the derivative of the inner function.The derivative of \( 5\ln(x^3 + 1) \) is \( 5 \cdot \frac{1}{x^3 + 1} \cdot (3x^2) \).

Step 4: Simplify the Derivative

The derivative becomes \[ f'(x) = \frac{5 \cdot 3x^2}{x^3 + 1} = \frac{15x^2}{x^3 + 1} \] after simplifying.

Key Concepts

Chain Rule

The chain rule is a powerful tool in calculus for differentiating composite functions. It helps when you have a function inside another function, and you need to find the derivative. In simpler terms, think of it as peeling an onion, where each layer represents a different part of the function. To use the chain rule, follow these basic steps:

• Identify the outer function and the inner function.
• Dabble the derivative of the outer function with respect to the inner function.
• Multiply this by the derivative of the inner function with respect to the variable.

In the exercise you are tackling, the chain rule makes it possible to break down the derivative into manageable parts. When you identify the function as a composite, it's easier to apply the chain rule methodically, focusing on each differentiation step separately. Remember, practice makes perfect when mastering techniques like the chain rule.

Composite Function

Understanding what a composite function is will greatly help when tackling problems that need the chain rule. A composite function is a function made up of two or more functions. This means when you have a function like \( f(g(x)) \), it consists of an outer function \( f \) and an inner function \( g \). For example, in the exercise, \( f(x) = \ln((x^3 + 1)^5) \) is a composite function. The outer function is the logarithm, while the inner part \( (x^3 + 1)^5 \) forms the core. This layered structure requires careful differentiation, accounting for each component with the appropriate rules. Decomposing a complex function into these simpler parts allows us to tackle them systematically. Once you identify each part, you can easily differentiate step by step using rules like the chain rule, ultimately simplifying the process.

Properties of Logarithms

Logarithms have special properties that can simplify mathematical expressions and make differentiation easier. One such property is the power rule for logarithms: \( \ln(a^b) = b\ln(a) \). This rule allows you to bring the exponent down and turn a power into a multiplication, making differentiation much simpler. In the exercise, we used this property to reformulate \( \ln((x^3 + 1)^5) \) into \( 5\ln(x^3 + 1) \). This dramatic simplification turns a potentially complex problem into something manageable. By rewriting expressions using these logarithm properties, you can remove complicating factors, allowing you to focus on the primary components of the function. Recognizing how and when to apply these rules is key to streamlining the differentiation process and achieving correct results.