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UPSC NDA Math Model Paper 1

Question : 40 of 120

Marks: +1, -0

Differentiate

x^{x^{x}}

with respect to 'x'.

Solution:

Consider,

f(x)=x^{x^{x}}

Taking log on both side, we get

\Rightarrow \log f(x)=\log x^{x^{x}}

\Rightarrow \log f(x)=x^{x} \log x

Now, use the Product Rule for Differentiation, we get

\frac{d}{dx}(uv)=u \frac{dv}{dx}+v \frac{du}{dx}

\Rightarrow \frac{f'(x)}{f(x)}=(x^{x}) \frac{1}{x}+\log x \cdot [x^{x}(1+\log x)]

\Rightarrow \frac{f'(x)}{f(x)}=(x^{x})\left[\frac{1}{x}+\log x \cdot (1+\log x)\right]

\Rightarrow f'(x)=f(x) \cdot (x^{x})\left[\frac{1}{x}+\log x \cdot (1+\log x)\right]

\Rightarrow f'(x)=x^{x^{x}} \cdot (x^{x})\left[\frac{1}{x}+\log x \cdot (1+\log x)\right]

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