TS EAMCET · Maths · Differentiation
If \(\mathrm{e}^{\mathrm{x}}=\mathrm{y}+\sqrt{y^2-1}\), then \(\frac{d y}{d x}=\)
- A \(\sinh x\)
- B \(\cosh x\)
- C \(\tanh x\)
- D \(-\sinh x\)
Answer & Solution
Correct Answer
(A) \(\sinh x\)
Step-by-step Solution
Detailed explanation
\[ \begin{aligned} & e^x=y+\sqrt{y^2-1} \\ \Rightarrow & e^x-y=\sqrt{y^2-1} \Rightarrow\left(e^x-y\right)^2=y^2-1 \\ \Rightarrow & e^{2 x}+y^2-2 y e^x=y^2-1 \\ \Rightarrow & e^{2 x}-2 e^x \cdot y=-1\left(\because y=\frac{e^{2 x}+1}{2 e^x}=\cos h x\right) \end{aligned} \]…
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