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UPSC NDA I 2021 Mathematics Solved Paper

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Question : 76 of 120

Marks: +1, -0

What is the derivative of

e^{x}

with respect to

x^{e}

?

$dfrac{x e^x}{e x^e}$
$dfrac{e^x}{x^e}$
$dfrac{x e^x}{x^e}$
$dfrac{e^x}{e x^e}$

Solution:

Concept:

The derivative of

f(x)

with respect to

g(x)

is

\frac{f'(x)}{g'(x)}

, provided both derivatives exist.

Explanation:

Let

f(x) = e^{x}

and

g(x) = x^{e}

.

Compute

f'(x) = \frac{d}{dx}e^{x} = e^{x}

.

Compute

g'(x) = \frac{d}{dx}x^{e} = e x^{e-1}

.

The required derivative is

\frac{f'(x)}{g'(x)} = \frac{e^{x}}{e x^{e-1}}

.

Multiply numerator and denominator by

x

:

\frac{x e^{x}}{e \cdot x \cdot x^{e-1}} = \frac{x e^{x}}{e x^{e}}

.

Thus, the derivative is

\frac{x e^{x}}{e x^{e}}

.

Answer:

\frac{x e^{x}}{e x^{e}}

(Option A)

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