Question

Find: \[ \frac{dy}{dx}, \quad \text{if} \quad y = x \tan x + \frac{\sqrt{x^2 + 1}}{2} \]

Hint:

When differentiating expressions involving trigonometric functions, don't forget to apply the product rule for terms like \( x \tan x \), and use the chain rule for composite functions like \( \frac{\sqrt{x^2 + 1}}{2} \).

Answer 1

Given \( y = x \tan x + \frac{\sqrt{x^2 + 1}}{2} \), differentiate with respect to \( x \). Apply the product rule to the first term and the chain rule to the second term:\[ \frac{dy}{dx} = \frac{d}{dx} \left( x \tan x \right) + \frac{d}{dx} \left( \frac{\sqrt{x^2 + 1}}{2} \right)\]After applying the rules, the derivative is:\[ \frac{dy}{dx} = \tan x + x \sec^2 x + \frac{x}{\sqrt{x^2 + 1}}\]