Question

Find derivative of (x)^{log x} with respect to x
1. \(\rm x^{x-1} (\log x^2)\)
2. \(\rm x^{\log x-1} (\log x^2)\)
3. \(\rm x^{log \;x} (\log x^2)\)
4. ​None of these​

Answer 1

Correct Answer - Option 2 : \(\rm x^{\log x-1} (\log x^2)\)

Concept:

Formula:

log mn = n log m

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

\(\rm \frac{d\log x}{dx} = \frac{1}{x}\)

\(\rm \frac{dx}{dx} = 1\)

Calculation:

Let y = x^log x

Taking log both sides, we get

⇒ log y = x^{log x}

⇒ log y = log x log  x            (∵ log mn = n log m)

Differentiating with respect to x, we get

⇒ \(\rm \frac{1}{y}\frac{dy}{dx} = \log x \frac{dlogx}{dx} + logx \frac{d\log x}{dx}\)

⇒ \(\rm \frac{dy}{dx} = y \left(\log x \times \frac{1}{x} + logx \times \frac{1}{x} \right )\)

⇒ \(\rm \frac{dy}{dx} = \frac{x^{\log x}}{x}\ (2\log x)\)

⇒ \(\rm \frac{dy}{dx} = x^{\log x-1} (\log x^2)\)