Problem 54 - Calculus of a Single Variable

Question

Find the derivative of the function. \(y=x \arctan 2 x-\frac{1}{4} \ln \left(1+4 x^{2}\right)\)

Step-by-Step Solution

Answer

The derivative of the function \(y=x \arctan 2 x-\frac{1}{4} \ln \left(1+4 x^{2}\right)\) is \(\frac{dy}{dx} = \arctan 2x\).

Step 1: Identify Each Term and its Corresponding Differentiation Rule

The function is composed of two terms: the first term, \(x \arctan 2x\), requires the product rule and chain rule for differentiation, while the second term, \(-\frac{1}{4} \ln (1+4x^2)\) requires the chain rule.

Step 2: Apply the Product Rule and Chain Rule on First Term

Differentiating the first term using the product rule, the derivative of \(x\) is 1 and the derivative of \(\arctan 2x\) using the chain rule will be \(\frac{2}{1 + 4x^2}\), so the derivative is \(\arctan 2x + \frac{2x}{1+4x^2}\).

Step 3: Apply Chain Rule on Second Term

Differentiating the second term using the chain rule, the derivative of \(-\frac{1}{4}\ln(1+4x^2)\) is \(-\frac{2x}{1 + 4x^2}\).

Step 4: Combine the Results

Combining the results of the individual differentiations will give you the final derivative of the given function.