Q105 / 160 · TS EAMCET 2023 (14 May Shift 1) · Mathematics

If $f(x)=e^x, h(x)=($ fof $)(x)$, then $\frac{h^{\prime}(x)}{h(x)}=$

  • A
    $h(x)$
  • B
    $\frac{1}{\mathrm{~h}(\mathrm{x})}$
  • Correct answer
    C
    $\log h(x)$
  • D
    $-\log \mathrm{h}(\mathrm{x})$

Solution

$f(x)=e^x$ $\begin{aligned} & h(x)=f(f(x))=e^{e^x} \\ & h^{\prime}(x)=e^{e^x} \cdot e^x=h(x) \cdot e^x \Rightarrow \frac{h^{\prime}(x)}{h(x)}=e^x \\ & \log h(x)=\log e^{e^x}=e^x \cdot \log e=e^x \\ & \therefore \frac{h^{\prime}(x)}{h(x)}=\log h(x)\end{aligned}$

Hence, the correct answer is option: 'C'.

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