Question Detail
Question
Find dy/dx if y = xtan x.
Correct Answer
Option D is the correct answer.
Detailed Explanation
To differentiate y = xtan x, we use logarithmic differentiation. Take the natural logarithm of both sides: ln y = ln(xtan x) = tan x · ln x. Now, differentiate both sides with respect to x using the product rule on the right side: (1/y) dy/dx = (d/dx(tan x)) · ln x + tan x · (d/dx(ln x))(1/y) dy/dx = (sec2 x) · ln x + tan x · (1/x)Multiply both sides by y: dy/dx = y · (sec2 x ln x + tan x / x)Substitute back y = xtan x:dy/dx = xtan x (tan x / x + sec2 x ln x).
Hint
Use logarithmic differentiation for functions of the form f(x)g(x).