If f'(1) = 2 and y=f( _ex), find d dx at x=e. - Vidyadip

Question

If f'(1) = 2 and $\text{y}=\text{f}(\log_\text{e}\text{x}),$ find $\frac{\text{d}}{\text{dx}}\text{at x}=\text{e}.$

✓

Answer

We have, f'(1) = 2 and $\text{y}=\text{f}(\log_\text{e}\text{x})$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\text{f}'(\log_\text{e}\text{x})\times\frac{\text{d}}{\text{dx}}(\log_\text{e}\text{x})$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{f}'(\log_\text{e}\text{x})\big(\frac{1}{\text{x}}\big)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{f}'(\log_\text{e}\text{e})\big(\frac{1}{\text{e}}\big) \big[\because\text{x}=\text{e}\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{f}'(1)\big(\frac{1}{\text{e}}\big) \big[\because\log_\text{e}\text{e}=1\big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{2}{\text{e}} \big[\because\text{f}'(1)=2\big]$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "2 Marks Questions" from Differentiation→Differentiation — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find a solution of $\frac{{dy}}{{dx}} = y\tan x$ which satisfy the condition y = 1 when x = 0.

Integrate the function: $\frac{1}{9 x^2+6 x+5}$

Prove the following Exercise:
$\int^{1}_{0}\text{x e}^{\text{x}}\ \text{dx}=1$

If f is an integrable function, show that:
$\int\limits^{\text{a}}_{-\text{a}}\text{f}\big(\text{x}^2\big)\text{dx}=2\int\limits^\text{a}_0\text{f}\big(\text{x}^2\big)\text{dx}$

The probability distribution of random variable X is given below:

$\text{X}$	$0$	$1$	$2$	$3$
$\text{P}(\text{X})$	$\text{k}$	$\frac{\text{k}}{2}$	$\frac{\text{k}}{4}$	$\frac{\text{k}}{8}$

Determine the value of k.

Find $\frac{dy}{dx}$ of the function $x^y + y^x = 1$

Write the unit vector in the direction of $\vec{\text{a}}=3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}$.

Evaluate the following determinant:
$\begin{vmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta \end{vmatrix}$

Show that $\text{f}(\text{x})=\log_{\text{a}}\text{x},0<\text{a}<1$ is a decreasing function for all x > 0.

Let $\text{A}=\begin{bmatrix}2&4\\3&2\end{bmatrix},\text{B}=\begin{bmatrix}1&3\\-2&5\end{bmatrix}$ and $\text{C}=\begin{bmatrix}-2&5\\3&4\end{bmatrix}.$ Find each of the following:
$3\text{A}-2\text{B}+3\text{C}$