KCET · Maths · Differentiation
If \(y=2^{\log x}\), then \(\frac{d y}{d x}\) is
- A \(\frac{2^{\log x}}{\log 2}\)
- B \(2^{\log x} \cdot \log 2\)
- C \(\frac{2^{\log x}}{x}\)
- D \(\frac{2^{\log x} \cdot \log 2}{x}\)
Answer & Solution
Correct Answer
(D) \(\frac{2^{\log x} \cdot \log 2}{x}\)
Step-by-step Solution
Detailed explanation
Given, \(y=2^{\log x}\)
\[
\begin{aligned}
&\Rightarrow \quad \frac{d y}{d x}=2^{\log x} \cdot \log _{e} 2 \cdot \frac{1}{x} \quad\left[\because \frac{d}{d x}\left(a^{x}\right)=a^{x} \log _{e} a\right] \\
&\Rightarrow \quad \frac{d y}{d x}=\frac{2^{\log x} \cdot \log _{e} 2}{x}
\end{aligned}
\]
\[
\begin{aligned}
&\Rightarrow \quad \frac{d y}{d x}=2^{\log x} \cdot \log _{e} 2 \cdot \frac{1}{x} \quad\left[\because \frac{d}{d x}\left(a^{x}\right)=a^{x} \log _{e} a\right] \\
&\Rightarrow \quad \frac{d y}{d x}=\frac{2^{\log x} \cdot \log _{e} 2}{x}
\end{aligned}
\]
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