Problem 51 - Calculus of a Single Variable

Question

Find the derivative of the function. \(y=\frac{1}{2}\left(\frac{1}{2} \ln \frac{x+1}{x-1}+\arctan x\right)\)

Step-by-Step Solution

Answer

The derivative of the function is \(y'=\frac{1}{4(x^2 - 1)} + \frac{1}{2(1+x^2)}\).

Step 1: Identify the functions

In the given function \(y=\frac{1}{2}\left(\frac{1}{2} \ln \frac{x+1}{x-1}+\arctan x\right)\), two functions are embedded: \(\ln \frac{x+1}{x-1}\) and \(\arctan x\).

Step 2: Apply the chain rule and the rules of logarithmic derivatives

The chain rule states that the derivative of a composition of functions is the derivative of the outer function multiplied by the derivative of the inner function. The derivative of \(\ln \frac{x+1}{x-1}\) is \(\frac{1}{\frac{x+1}{x-1}} \cdot \frac{d}{dx} (\frac{x+1}{x-1})\). This simplifies by applying the quotient rule to \(\frac{d}{dx} (\frac{x+1}{x-1})\), resulting in \(\frac{1}{x^2 - 1}\).

Step 3: Find the derivative of \(\arctan x\)

The derivative of \(\arctan x\) is \(\frac{1}{1+x^2}\).

Step 4: Combine the results

The derivative of the total function will be the sum of the derivatives for each of the separate functions: \(\frac{1}{4} \cdot \frac{1}{x^2 - 1} + \frac{1}{2} \cdot \frac{1}{1+x^2}\).

Step 5: Simplify the result

Simplifying the result yields \(y'=\frac{1}{4(x^2 - 1)} + \frac{1}{2(1+x^2)}\).