Problem 39 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. $$ f(x)=\ln \frac{1-x}{1+2 x} $$

Step-by-Step Solution

Answer

The derivative is \( \frac{-3(1+2x)}{(1-x)(1+2x)^2} \).

Step 1: Understand the Function

The given function is written as a natural logarithm of a quotient: \( \ln \left( \frac{1-x}{1+2x} \right) \). To differentiate this function, we can use the chain rule along with the logarithmic differentiation rules.

Step 2: Apply the Logarithm Rule

Recall that the derivative of \( \ln(u) \) with respect to \( x \), where \( u \) is a function of \( x \), is \( \frac{1}{u} \frac{du}{dx} \). Here, \( u = \frac{1-x}{1+2x} \). So, we first need to find \( \frac{d}{dx} \left( \ln \left( \frac{1-x}{1+2x} \right) \right) \).

Step 3: Differentiate the Argument with Respect to x

Differentiate the quotient \( \frac{1-x}{1+2x} \) using the quotient rule, which states \( \frac{d}{dx} \left( \frac{v}{w} \right) = \frac{w \cdot \frac{dv}{dx} - v \cdot \frac{dw}{dx}}{w^2} \). Here, \( v = 1-x \) and \( w = 1+2x \).

Step 4: Differentiate the Numerator and Denominator

Calculate \( \frac{dv}{dx} = -1 \) and \( \frac{dw}{dx} = 2 \).

Step 5: Apply the Quotient Rule

Substitute the derivatives into the quotient rule formula: \( \frac{d}{dx} \left( \frac{1-x}{1+2x} \right) = \frac{(1+2x)(-1) - (1-x)\cdot2}{(1+2x)^2} \). Simplify this expression.

Step 6: Simplify the Derivative of the Argument

Simplify the numerator: \( (-1-2x - 2+2x) \), which simplifies to \( -3 \). Thus, \( \frac{d}{dx} \left( \frac{1-x}{1+2x} \right) = \frac{-3}{(1+2x)^2} \).

Step 7: Apply the Chain Rule

Use the chain rule for the logarithmic derivative: \( \frac{d}{dx} \left( \ln(u) \right) = \frac{1}{u} \cdot \frac{du}{dx} \). Substituting the values, we get \( \frac{d}{dx} \left( \ln \left( \frac{1-x}{1+2x} \right) \right) = \frac{1}{\frac{1-x}{1+2x}} \cdot \frac{-3}{(1+2x)^2} \).

Step 8: Simplify the Final Expression

Simplify \( \frac{1}{\frac{1-x}{1+2x}} \) to \( \frac{1+2x}{1-x} \). Therefore, the final derivative is \( \frac{-3(1+2x)}{(1-x)(1+2x)^2} \). Simplify further, if needed, for a concise expression.

Key Concepts

Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. When you have a function inside another function, like \( f(g(x)) \), the chain rule comes into play. To differentiate such a function, you need to follow these steps:

• Differentiate the outer function with respect to the inner function.
• Multiply this result by the derivative of the inner function with respect to \( x \).

This essentially means "sweeping through the layers" of the function, processing from the outside in.

To illustrate using our example: we have \( f(x) = \ln(u) \), where \( u = \frac{1-x}{1+2x} \). The derivative of \( \ln(u) \) is \( \frac{1}{u} \), which is the outer function, and then you multiply it by the derivative of \( u \) (obtained via quotient rule), the inner function. This is how the chain rule helps in nesting derivatives.

Quotient Rule

The quotient rule is used when differentiating a fraction or a ratio of two functions. This is crucial whenever the function you want to differentiate is of the form \( \frac{v}{w} \), where both \( v \) and \( w \) are functions of \( x \). The rule states:

• Take the derivative of the numerator \( v \), multiply it by the denominator \( w \).
• Subtract the product of the numerator \( v \) and the derivative of the denominator \( w \).
• Divide the entire expression by the square of the denominator \( w^2 \).

This translates to the formula \( \frac{d}{dx} \left( \frac{v}{w} \right) = \frac{w \cdot \frac{dv}{dx} - v \cdot \frac{dw}{dx}}{w^2} \).

Applying this to our expression within the logs, \( \frac{1-x}{1+2x} \), we found: \( \frac{d}{dx} \left( \frac{1-x}{1+2x} \right) = \frac{-3}{(1+2x)^2} \). This derivatives the building block needed to tackle the logarithmic differentiation.

Logarithmic Differentiation

Logarithmic differentiation is a powerful method that simplifies the process of differentiating functions by utilizing the properties of logarithms. Especially handy with functions presented as products, quotients, or powers, it helps simplify the differentiation process.

• Write the function in a logarithmic form.
• Differentiate this form using the chain rule combined with the basic derivative of a logarithm.
• Revert back to non-logarithmic form if required.

In the context of the given function, \( \ln\left( \frac{1-x}{1+2x} \right) \) helps break down complexity into a manageable form.

The steps involve first leveraging the quotient property of logs and then employing the chain and quotient rules. For instance, the derivative \( \frac{d}{dx} \left( \ln \left( \frac{1-x}{1+2x} \right) \right) = \frac{-3(1+2x)}{(1-x)(1+2x)^2} \) is simplified through careful application of logarithmic differentiation principles.