TS EAMCET · Maths · Differentiation
If \(f(x)=e^x, h(x)=(\) fof \()(x)\), then \(\frac{h^{\prime}(x)}{h(x)}=\)
- A \(h(x)\)
- B \(\frac{1}{\mathrm{~h}(\mathrm{x})}\)
- C \(\log h(x)\)
- D \(-\log \mathrm{h}(\mathrm{x})\)
Answer & Solution
Correct Answer
(C) \(\log h(x)\)
Step-by-step Solution
Detailed explanation
\(f(x)=e^x\) \(\begin{aligned} & h(x)=f(f(x))=e^{e^x} \\ & h^{\prime}(x)=e^{e^x} \cdot e^x=h(x) \cdot e^x \Rightarrow \frac{h^{\prime}(x)}{h(x)}=e^x \\ & \log h(x)=\log e^{e^x}=e^x \cdot \log e=e^x \\ & \therefore \frac{h^{\prime}(x)}{h(x)}=\log h(x)\end{aligned}\)
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