Q.37 If y = xx, then dy/dx is
- xx(x − 1)
- xx−1
- xx(1 + log x)
- ex(1 + log x)
Introduction
Finding the derivative of y = xx is a standard calculus problem.
Since both the base and exponent depend on x, normal power rules cannot be applied.
We solve it using logarithmic differentiation.
Given Question
If
y = xx
then find
dy/dx
Options
- (A) xx(x − 1)
- (B) xx−1
- (C) xx(1 + log x)
- (D) ex(1 + log x)
Step-by-Step Solution
Step 1: Take Natural Logarithm
y = xx log y = log(xx) = x log x
Step 2: Differentiate Both Sides
(1 / y) · (dy / dx) = d/dx (x log x) = log x + 1
Step 3: Multiply by y
dy/dx = y (log x + 1)
dy/dx = xx(1 + log x)
Correct Answer
Option (C):
xx(1 + log x)
Final Result
The derivative of y = xx is:
dy/dx = xx(1 + log x)
