MHT CET · Maths · Differentiation
If \(x=e^{\left(\frac{x}{y}\right)}\), then \(\frac{d y}{d x}=\)
- A \(\frac{x-y}{x \log y}\)
- B \(\frac{x-y}{y \log x}\)
- C \(\frac{x-y}{x \log x}\)
- D \(\frac{x+y}{x \log x}\)
Answer & Solution
Correct Answer
(C) \(\frac{x-y}{x \log x}\)
Step-by-step Solution
Detailed explanation
\(x=e^{\left(\frac{x}{y}\right)} \Rightarrow \log _e x=\frac{x}{y} \Rightarrow y \log _e x=x\)
Differentiating both sides w.r.t \(x\) we get \(\frac{d y}{d x} \cdot \log _e x+y \cdot \frac{1}{x}=1\)
\(\Rightarrow \frac{d y}{d x} \cdot x \log _e x+y=x\)
\(\Rightarrow \frac{d y}{d x}=\frac{x-y}{x \log _e x}\)
Change the option (C) as \(\frac{x-y}{x \log x}\)
Differentiating both sides w.r.t \(x\) we get \(\frac{d y}{d x} \cdot \log _e x+y \cdot \frac{1}{x}=1\)
\(\Rightarrow \frac{d y}{d x} \cdot x \log _e x+y=x\)
\(\Rightarrow \frac{d y}{d x}=\frac{x-y}{x \log _e x}\)
Change the option (C) as \(\frac{x-y}{x \log x}\)
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