Problem 44 - Calculus of a Single Variable

Question

Find the derivative of the function. \(f(x)=\arctan \sqrt{x}\)

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=\arctan \sqrt{x}\) is \(f'(x) = \frac{1}{2(x+1)x^{1/2}}\)

Step 1: Identify the expression for differentiation

Our main task is to differentiate the function \(f(x) = \arctan \sqrt{x}\) this is a composition of functions. First, we have \(\sqrt{x}\) and then \(\arctan \(x)\). We'll begin by differentiating the outermost function using the chain rule.

Step 2: Apply the chain rule

The chain rule states that the derivative of a composite function is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. By applying chain rule, we have \(f'(x) = \frac{1}{1 + (\sqrt{x})^2} * \frac{1}{2}x^{-1/2}\) as the derivative of our original function.

Step 3: Simplify the result

We can simplify our derived function to obtain \(f'(x) = \frac{1}{2(x+1)x^{1/2}}\) by multiplying out our previous result.