Match List-I with List-II

List-I	List-II
(Function)	(Derivative with respect to 'x')
(A) $f(x) = x^x$	(I) $ax^{a-1}$
(B) $f(x) = a^x$	(II) 0
(C) $f(x) = a^a$	(III) $a^x log_e a$
(D) $f(x) = x^a$	(IV) $x^x(1 + log_e x)$

Choose the correct answer from the options given below:

$y = x^x$

$\log y = x \log x$

Differentiating both sides w.r.t. $x$:

$\frac{1}{y} \cdot \frac{dy}{dx} = x \cdot \frac{1}{x} + \log x \cdot 1$

$\frac{1}{y} \cdot \frac{dy}{dx} = 1 + \log x$

$\frac{dy}{dx} = y(1 + \log x)$

$\boxed{\frac{dy}{dx} = x^x(1 + \log_e x)}$ → (IV)

$f(x) = a^x$

$\boxed{\frac{dy}{dx} = a^x \log_e a}$ → (III)

$f(x) = a^a$

Here $a$ is a constant, so $a^a$ is also a constant.

$\boxed{\frac{dy}{dx} = 0}$ → (II)

$f(x) = x^a$

Using power rule with $n = a$ (constant):

$\boxed{\frac{dy}{dx} = a \cdot x^{a-1}}$ → (I)

(A) → (IV), (B) → (III), (C) → (II), (D) → (I)

Option 3 is correct ✓