MHT CET · Maths · Differentiation
If \(y=\sqrt{e^{\sqrt{x}}}\), then \(\frac{d y}{d x}=\)
- A \(\frac{e^{\sqrt{x}}}{4 \sqrt{x}}\)
- B \(\frac{e^{\sqrt{x}}}{4 x}\)
- C \(\frac{e^{\frac{\sqrt{x}}{2}}}{4 \sqrt{x}}\)
- D \(\frac{\mathrm{e}^{\sqrt{\mathrm{x}}}}{2 \sqrt{\mathrm{x}}}\)
Answer & Solution
Correct Answer
(C) \(\frac{e^{\frac{\sqrt{x}}{2}}}{4 \sqrt{x}}\)
Step-by-step Solution
Detailed explanation
\(
\begin{aligned}
& 2 \log y=\sqrt{x} \log e \Rightarrow 2 \log y=\sqrt{x} \\
& \frac{2}{y} \frac{d y}{d x}=\frac{1}{2 \sqrt{x}} \Rightarrow \frac{d y}{d x}=y\left[\frac{1}{4 \sqrt{x}}\right]=\frac{\sqrt{e^{\sqrt{x}}}}{4 \sqrt{x}}
\end{aligned}
\)
Taking \(\log\) on both sides,
\(
2 \log \mathrm{y}=\sqrt{\mathrm{x}} \log \mathrm{e} \Rightarrow 2 \log \mathrm{y}=\sqrt{\mathrm{x}}
\)
Differentiating both sides w.r.t. \(x\), we get
\(
\frac{2}{y} \frac{d y}{d x}=\frac{1}{2 \sqrt{x}} \Rightarrow \frac{d y}{d x}=y\left[\frac{1}{4 \sqrt{x}}\right]=\frac{\sqrt{e^{\sqrt{x}}}}{4 \sqrt{x}}
\)
\begin{aligned}
& 2 \log y=\sqrt{x} \log e \Rightarrow 2 \log y=\sqrt{x} \\
& \frac{2}{y} \frac{d y}{d x}=\frac{1}{2 \sqrt{x}} \Rightarrow \frac{d y}{d x}=y\left[\frac{1}{4 \sqrt{x}}\right]=\frac{\sqrt{e^{\sqrt{x}}}}{4 \sqrt{x}}
\end{aligned}
\)
Taking \(\log\) on both sides,
\(
2 \log \mathrm{y}=\sqrt{\mathrm{x}} \log \mathrm{e} \Rightarrow 2 \log \mathrm{y}=\sqrt{\mathrm{x}}
\)
Differentiating both sides w.r.t. \(x\), we get
\(
\frac{2}{y} \frac{d y}{d x}=\frac{1}{2 \sqrt{x}} \Rightarrow \frac{d y}{d x}=y\left[\frac{1}{4 \sqrt{x}}\right]=\frac{\sqrt{e^{\sqrt{x}}}}{4 \sqrt{x}}
\)
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