Differentiate the following w.r.t.x: e^ x , x>0 - Vidyadip

Question

Differentiate the following w.r.t.x:

$\sqrt{\text{e}^\sqrt{\text{x}}},\ \text{x}>0$

✓

Answer

$\text{Let}\ \sqrt{\text{e}^\sqrt{\text{x}}}=(\sqrt{\text{e}^\sqrt{\text{x}}})^{\frac{1}{2}}$
$\therefore\ \frac{\text{dy}}{\text{dx}} =\frac{{1}}{{2}} (\sqrt{\text{e}^\sqrt{\text{x}}})^{\frac{1}{2}}.\frac{\text{d}}{\text{dx}}(\sqrt{\text{e}^\sqrt{\text{x}}})=\frac{1}{2\sqrt{\text{e}^\sqrt{\text{x}}}}.\sqrt{\text{e}^\sqrt{\text{x}}}\frac{\text{d}}{\text{dx}}(\sqrt{\text{x}})$
$=\frac{1}{2\sqrt{\text{e}^\sqrt{\text{x}}}}.\text{e}\sqrt{\text{x}}.\frac{\text{1}}{{2\sqrt{\text{x}}}}=\frac{1}{4}\frac{{\text{e}^\sqrt{\text{x}}}}{\sqrt{\text{x e}^\sqrt{\text{x}}}}$

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" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate $\int\frac{\text{e}^{\tan^{-1\text{x}}}}{1+\text{x}^2}\text{ dx}$

Find the minor of element 6 in the determinant $\Delta=\left|\begin{array}{lll} {1} & {2} & {3} \\ {4} & {5} & {6} \\ {7} & {8} & {9} \end{array}\right|$

Find the value of $\lambda$ so that the points (1, - 5), (-4, 5) and $(\lambda,7)$ are collinear.

If X follows binomial distribution with parameters n = 5, p and P(X = 2) = 9P(X = 3), then find the value of p.

If a line makes angle $\alpha, \beta, \gamma$ with the coordinate axes, then find the value of $\cos 2 \alpha+\cos 2 \beta+$ $\cos 2 \gamma$.

Compute the indicated products. $\left[\begin{array}{cc} {2} & {1} \\ {3} & {2} \\ {-1} & {1} \end{array}\right]\left[\begin{array}{ccc} {1} & {0} & {1} \\ {-1} & {2} & {1} \end{array}\right]$

Write the value of $\cot^{-1}(-\text{x})$ for all $\text{x}\in\text{R}$ in terms of $\cot^{-1}\text{x}$

Find the area of the region enclosed by $x$-axis, the curve $y=\sin ^3 x \cos x$ and the ordinates $x=0$ and $x=\frac{\pi}{2}$.

If $\text{f(x)}=\begin{cases}\frac{\text{x}^2-16}{\text{x}-4},&\text{if }\text{ x}\neq4\\\text{k},&\text{if }\text{ x}=4\end{cases}$ is continuous at x = 4, find k.

State whether the function is one-one, onto or bijective. Justify your answer. f: R $\rightarrow$ R defined by f(x) = 3 - 4x.