Question

If $y=2^{\log x}$, then$\frac{dy}{dx}$ is equal to

• A. $\frac{2^{\log x}}{\log 2}$
• B. $2^{\log x}\times \log 2$
• C. $\frac{2^{\log x}}{x}$
• D. $\frac{2^{\log x}\times \log 2}{x}$

Answer 1

The correct option is D

$\frac{2^{\log x}\times \log 2}{x}$

Explanation for the correct answer:

$y=2^{\log x}$

Taking logarithm on both sides we get

$\log y=\log 2^{\log x}$

$\Rightarrow$$\log y=\log x\times \log 2$ $\mathbf{.}\mathbf{.}\mathbf{.}\left[\because \log a^m=m\times \log a\right]$

Differentiating with respect to $x$ on both sides we get

$\Rightarrow$$\frac{1}{y}\times \frac{dy}{dx}=\log 2\left(\frac{1}{x}\right)$

$\Rightarrow$ $\frac{dy}{dx}=y\times \log 2\left(\frac{1}{x}\right)$

Substituting the value of $y=2^{\log x}$ we get

$\Rightarrow$ $\frac{dy}{dx}=\frac{2^{\log x}\times \log 2}{x}$

Hence, option (D) is the correct option.