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Differential Calculus

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Question : 27 of 69

Marks: +1, -0

Differentiate

e^{(x^{x})}

$(1+\log x)$
$x^{x}(1+\log x)$
$e^{x^{x}}(1+\log x) x^{x}$
$e^{x^{x}}(1+\log x)$

Solution:

Concept:

Differentiate using chain rule and logarithmic differentiation for

x^x

.

Explanation:

Let

y = e^{(x^x)}

.

Set

u = x^x

, then

y = e^u

, so

\frac{dy}{dx} = e^u \cdot \frac{du}{dx}

.

To find

\frac{du}{dx}

, take

\ln u = x \ln x

.

Differentiate:

\frac{1}{u} \frac{du}{dx} = \ln x + 1

.

Thus

\frac{du}{dx} = u (1 + \ln x) = x^x (1 + \ln x)

.

Substitute back:

\frac{dy}{dx} = e^{x^x} \cdot x^x (1 + \ln x)

.

Answer:

e^{x^x} (1 + \log x) x^x

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