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If f(1)=5 ; f^{\prime}(1)=3  find the value of the derivative of \log f\left(e^x\right) w.r.t. x at the point x=0. .

Option: 1

\frac{6}{5}


Option: 2

\frac{7}{3}


Option: 3

\frac{1}{3}


Option: 4

\frac{3}{5}


Answers (1)

best_answer

Let y=\log f\left(e^x\right)

Then take derivative of the given term,

\begin{aligned} & \frac{d y}{d x}=\frac{1}{f\left(e^x\right)} \cdot \frac{d}{d x}\left(f\left(e^x\right)\right) \\ &=\frac{1}{f\left(e^x\right)} \cdot f^{\prime}\left(e^x\right) \cdot \frac{d}{d x}\left(e^x\right) \\ &=\frac{1}{f\left(e^x\right)} \cdot f^{\prime}\left(e^x\right) \cdot e^x \\ & \therefore \frac{d y}{d x} \quad=\frac{1}{f\left(e^0\right)} \cdot f^{\prime}\left(e^0\right) \cdot e^0 \\ &=\frac{1}{f(1)} \cdot f^{\prime}(1) \cdot 1 \\ &=\frac{f^{\prime}(1)}{f(1)} \end{aligned}

Now, Putting the value of f'(1) and f(1) in the obtained value of \frac{d y}{d x}{ }_{x=0,}

\begin{aligned} \frac{d y}{d x}_{x=0} & =\frac{f^{\prime}(1)}{f(1)} \\ & =\frac{3}{5} \end{aligned}

Posted by

Ritika Jonwal

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