Problem 55

Question

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\log \left(1-x^{2}\right) $$

Step-by-Step Solution

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Answer

The derivative of $ f(x) = \log(1-x^2) $ is $ \frac{-2x}{(1-x^2)\ln(10)} $.

1Step 1: Apply the Chain Rule

The function given is $ f(x) = \log(1-x^2) $. To find its derivative, we apply the chain rule. The logarithm base 10 derivative is $ \frac{1}{ ext{argument}} \cdot \text{ln}(10) $ times the derivative of the argument. Here, the argument is $ 1-x^2 $.

2Step 2: Differentiate the Inner Function

Differentiate the inner function $ 1-x^2 $ with respect to $ x $. The derivative of $ 1 $ is $ 0 $, and the derivative of $ -x^2 $ is $ -2x $. Therefore, the derivative of $ 1-x^2 $ is $ -2x $.

3Step 3: Find the Derivative of the Logarithm

Using the derivative of the logarithm, the derivative of $ \log(1-x^2) $ is $ \frac{1}{1-x^2} \cdot \frac{1}{\ln(10)} $, with the inner derivative included, resulting in $ \frac{-2x}{1-x^2} \cdot \frac{1}{\ln(10)} $.

4Step 4: Simplify the Expression

Multiply out the terms to find the full derivative: $ \frac{-2x}{(1-x^2)\ln(10)} $. This is the simplified derivative of the given function $ f(x) = \log(1-x^2) $.

Key Concepts

Chain RuleLogarithmic DifferentiationDerivative of a Function

Chain Rule

To differentiate composite functions, the chain rule is an essential tool in calculus. The chain rule allows you to find the derivative of a function that is nested within another function. Imagine you have a function where there's an outer expression and an inner expression (often referred to as the "argument"). To apply the chain rule, follow these steps:

First, differentiate the outer function, considering the inner function as a simple variable.
Then, multiply this result by the derivative of the inner function.

Using this rule ensures that all changes in the inner function affecting the outer function are accounted for in the derivative. For instance, if you have a function like $ f(x) = \, \operatorname{log}(1-x^2) $, the outer function is the logarithm and the inner is $ 1-x^2 $. Differentiate $ \log(u) $, where $ u = 1-x^2 $, and multiply by the derivative of $ u $, giving the final derivative result.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions involving logarithms, and it can simplify the differentiation process—especially when dealing with products or quotients of functions. When the logarithm is involved, the differentiation follows a specific pattern:

Take the derivative of the logarithmic function's outer layer by using the formula $ \frac{1}{\text{argument}} \cdot \ln(a) $, where $ a $ is the base of the logarithm.
Multiply that by the derivative of the inner function (the argument of the logarithm).

In the context of $ \operatorname{log}(1-x^2) $, you first take the derivative of $ \operatorname{log}(u) $ as $ \frac{1}{u} $ and adjust for the base with $ \ln(10) $, followed by the inner derivative $ -2x $. This results in the correct derivative when multiplied together.

Derivative of a Function

Calculating the derivative of a function is a fundamental skill in calculus that represents the rate of change of a function with respect to its variable. Here's a simple step-by-step method to find the derivative of any standard function:

Identify the function type: Is it elementary, composite, or involving powers or products?
Apply the appropriate differentiation rule: such as the power rule, product rule, or chain rule for composite functions.

The derivative of $ f(x) = \operatorname{log}(1-x^2) $ was found using a combination of the chain and logarithmic differentiation rules:

The derivative of the logarithm $ \operatorname{log}(u) $ involved $ \frac{1}{u \cdot \ln(10)} $.
The derivative of the inner function $ 1-x^2 $ was $ -2x $.
Combining these using the chain rule, you obtain: $ \frac{-2x}{(1-x^2)\ln(10)} $.

This provides a complete method of finding how a function's value changes with its input.

Problem 55

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