# What is the derivative of ex with respect to xe?

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

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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) = ex

f'(x) = $\rm {d\over dx}e^x = e^x$

g(x) = xe

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

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

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