Question

What is the derivative of e^x with respect to x^e?
1. \(\dfrac{xe^x}{ex^e}\)
2. \(\dfrac{e^x}{x^e}\)
3. \(\dfrac{xe^x}{x^e}\)
4. \(\dfrac{e^x}{ex^e}\)

Answer 1

Correct Answer - Option 1 : \(\dfrac{xe^x}{ex^e}\)

Concept:

The differentiation of f(x) wrt g(x) = \(\rm f'(x)\over g'(x)\)

f(x) and g(x) be the function of x

Calculation:

f(x) = e^x 

f'(x) = \(\rm {d\over dx}e^x = e^x\)

g(x) = x^e 

g'(x) = \(\rm {d\over dx}x^e = ex^{e-1}\)

Required derivative D = \(\rm f'(x)\over g'(x)\)

D = \(\rm e^x\over ex^{e-1}\) = \(\rm xe^x\over ex^e\)