Differentiate the following functions with respect to x: e^ x - Vidyadip

Question

Differentiate the following functions with respect to x:
$\text{e}^{\sin\sqrt{\text{x}}}$

✓

Answer

Let, $\text{y}=\text{e}^{\sin\sqrt{\text{x}}}$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\text{e}^{\sin\sqrt{\text{x}}}\big)$
$=\text{e}^{\sin\sqrt{\text{x}}}\frac{\text{d}}{\text{dx}}\big(\sin\sqrt{\text{x}}\big)$
[Using chain rule]
$=\text{e}^{\sin\sqrt{\text{x}}}\times\cos\sqrt{\text{x}}\frac{\text{d}}{\text{dx}}\sqrt{\text{x}}$
[Using chain rule]
$=\text{e}^{\sin\sqrt{\text{x}}}\times\cos\sqrt{\text{x}}\times\frac{1}{2\sqrt{\text{x}}}$
$=\frac{1}{2\sqrt{\text{x}}}\times\cos\sqrt{\text{x}}\times\text{e}^{\sin\sqrt{\text{x}}}$
So,
$\frac{\text{d}}{\text{dx}}=\big(\text{e}^{\sin^\sqrt{\text{x}}}\big)=\frac{1}{2\sqrt{\text{x}}}\cos\sqrt{\text{x}}\times\text{e}^{\sin\sqrt{\text{x}}}$

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 "3 Marks Question" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

An urn contain 5 red and 2 black balls. Two balls rendomly selected. Let X represent the number of black ball. What are the possible values of X. Is X a random variable?

Solve for x:
$2 \tan^{-1}(\cos x) = \tan^{-1} (2 cosec\ x)$.

Evaluate the following integrals:
$\int^\limits\frac{\pi}{4}_{0}\sin^32\text{t}\cos2\text{t}\text{ dt}$

Solve the following differential equations:
$\frac{\text{dy}}{\text{dx}}=\text{e}^{\text{x+y}}+\text{e}^{-\text{x+y}}$

Three relation $R_4$ is defined in set $A = \{a, b, c\}$ as follows:
$R_4 = \{(a, b), (b, c), (c, a)\}$
Find whether or not the relation $R_4$ on $A$ is:

Reflexive.
Symmetric.
Transitive.

Each of the following defines a relation on N:
x, y is square of an integer $\text{x},\text{y}\in\text{N}$
Determine which of the above relations are reflexive, symmetric and transitive.

Evaluate the following integrals:
$\int\cos^5\text{x}\text{ dx}$

Evalute the following integrals:
$\int\frac{\sec\text{x}}{\sec2\text{x}}\text{dx}$

If $\vec a = 2\hat i + 2\hat j + 3\hat k,\vec b = - \hat i + 2\hat j + \hat k$ and $\vec c = 3\hat i + \hat j$ are such that $\vec a + \lambda \vec b$ is perpendicular to $\vec c$, then find the value of $\lambda $

Find the shortest distance between lines $\vec{r}=6 \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k})$ and $\vec{r}=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k})$