Question Detail

To find the derivative of y = x^(tan x), we use logarithmic differentiation.1. Take the natural logarithm of both sides: ln y = ln(x^(tan x)) Using the logarithm property ln(a^b) = b ln(a): ln y = tan x * ln x2. Differentiate both sides implicitly with respect to x: d/dx(ln y) = d/dx(tan x * ln x) (1/y) dy/dx = (d/dx(tan x)) ln x + tan x (d/dx(ln x)) (using the product rule on the right side)3. Calculate the derivatives: d/dx(tan x) = sec² x d/dx(ln x) = 1/x4. Substitute these derivatives back: (1/y) dy/dx = (sec² x) ln x + tan x (1/x) (1/y) dy/dx = sec² x ln x + (tan x) / x5. Solve for dy/dx by multiplying both sides by y: dy/dx = y * (sec² x ln x + (tan x) / x)6. Substitute back y = x^(tan x): dy/dx = x^(tan x) * (sec² x ln x + (tan x) / x) Rearranging the terms inside the parenthesis: dy/dx = x^(tan x) * ((tan x) / x + sec² x ln x)Thus, option D is correct.