Problem 17 - Calculus Single Variable

Question

Calculate the derivative with respect to \(x\) of the given expression. \(\ln (\sqrt{1-x})\)

Step-by-Step Solution

Answer

The derivative is \( \frac{-1}{2(1-x)} \).

Step 1: Apply the chain rule to the logarithmic function

The function is given as \( \ln(\sqrt{1-x}) \). Let's rewrite it to make it easier to differentiate: \( \ln((1-x)^{1/2}) \). Using the chain rule for differentiation, the derivative of \( \ln(u) \) with respect to \(x\) is \( \frac{1}{u} \cdot \frac{du}{dx} \). Here, \(u = (1-x)^{1/2}\). So, first find \( \frac{d}{dx}[ (1-x)^{1/2} ] \).

Step 2: Differentiate the inner function using the power rule

Differentiate \( u = (1-x)^{1/2} \). The power rule states that \( \frac{d}{dx}[x^n] = n x^{n-1} \). Therefore, \( \frac{d}{dx}[(1-x)^{1/2}] = \frac{1}{2} (1-x)^{-1/2} \cdot \frac{d}{dx} (1-x) \). Now we need to find \( \frac{d}{dx} (1-x) = -1 \). Multiply these results together: \( \frac{1}{2} (1-x)^{-1/2} \cdot (-1) = -\frac{1}{2(1-x)^{1/2}} \).

Step 3: Apply the chain rule to complete the derivative

Substitute \( \frac{du}{dx} = -\frac{1}{2(1-x)^{1/2}} \) back into the derivative of \( \ln(u) \). So, \( \frac{d}{dx}[ \ln( (1-x)^{1/2} ) ] = \frac{1}{(1-x)^{1/2}} \cdot -\frac{1}{2(1-x)^{1/2}} \). Simplify this expression, \( = -\frac{1}{2} \cdot \frac{1}{1-x} = -\frac{1}{2(1-x)} \).

Step 4: Simplify the result

The final result for the derivative of \( \ln( \sqrt{1-x} ) \) with respect to \( x \) is \( \frac{-1}{2(1-x)} \). We simplify any additional fractions and ensure that the expression is in the simplest form.

Key Concepts

Logarithmic Differentiation

Logarithmic differentiation is a powerful technique that's particularly useful for differentiating complex expressions involving logarithms. Understanding the steps involved in this differentiation method can greatly aid your mathematical repertoire, especially when dealing with functions within a logarithm. When differentiating the natural logarithm of a function, such as \( \ln(u) \), with respect to \( x \), we need the chain rule and the derivative of \( \ln(u) \), which is \( \frac{1}{u} \). This means you multiply \( \frac{1}{u} \) by the derivative of \( u \) with respect to \( x \), noted as \( \frac{du}{dx} \). Doing so allows you to tackle the complexity of nested functions within logarithms, by breaking them down into simpler derivatives.
This technique enhances the efficiency and accuracy when working with intricate problems in calculus.

Chain Rule

The chain rule is an essential tool in calculus, useful for finding derivatives of compositions of functions. Think of it as a method to "unwrap" these functions, one layer at a time. In our problem, \( u = (1-x)^{1/2} \), where \( u(x) \) is nested inside another function. The chain rule allows us to differentiate \( \ln(u) \) by first taking the derivative of the logarithmic function, \( \frac{1}{u} \), and then multiplying it by \( \frac{du}{dx} \), the derivative of the inner function.
This logical assembly line ensures that each element of the composite function is appropriately accounted for in the differentiation process. Such a systematic approach helps simplify the differentiation of even the most complex composite functions.

Power Rule

The power rule is a straightforward and indispensable tool for differentiation involving powers of \( x \). For a function written in the form \( x^n \), where \( n \) is a real number, the power rule states that the derivative \( \frac{d}{dx}[x^n] = n x^{n-1} \). This tells us that we multiply by the exponent \( n \) and then reduce the exponent by one.
In our problem, the function \( (1-x)^{1/2} \) exemplifies how the power rule can quickly provide the derivative. Rewritten with the power rule, we obtain \( \frac{1}{2}(1-x)^{-1/2} \), and then multiply by the derivative of the inner function, \( \frac{d}{dx}(1-x) = -1 \).

• Apply the power rule effectively simplifies the differentiation process.
• It bypasses complex calculations allowing for a more direct pathway to the solution.

Simplifying Expressions

Simplifying expressions is the last crucial step in the differentiation process. After you've calculated the derivative, it's vital to ensure the expression is as succinct and straightforward as possible. This involves combining like terms, reducing fractions, and writing the expression in a form that's easier to interpret and use.
In this exercise, after calculating the derivative as \( \frac{d}{dx}[ \ln( (1-x)^{1/2} ) ] = \frac{1}{(1-x)^{1/2}} \cdot -\frac{1}{2(1-x)^{1/2}} \), we multiply and simplify it to \( -\frac{1}{2} \cdot \frac{1}{1-x} \). Once simplified, the expression becomes \( -\frac{1}{2(1-x)} \), a leaner and more accessible form.

• This step is essential for clear communication and understanding.
• Concise expressions are easier to apply in further mathematical work.