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

Question

Differentiate the following functions with respect to x:
$(\sin\text{x})^{\cos\text{x}}$

✓

Answer

Let $\text{y}=(\sin\text{x})^{\cos\text{x}}\ .....(\text{i})$
Taking log on both the sides,
$\log\text{y}=(\sin\text{x})^{\cos\text{x}}$
$\Rightarrow\log\text{y}=\cos\text{x}\log(\sin\text{x})$
Differentiating with respect to x,
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\cos\text{x}\frac{\text{d}}{\text{dx}}(\log\sin\text{x})+\log\sin\text{x}\frac{\text{d}}{\text{dx}}(\cos\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\cos\text{x}\frac{1}{\sin\text{x}}\frac{\text{d}}{\text{dx}}(\sin\text{x})+\log\sin\text{x}(-\sin\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\frac{\cos\text{x}}{\sin\text{x}}(\cos\text{x})-\sin\text{x}\log\sin\text{x}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}[\cos\text{x}\cot\text{x}-\sin\text{x}\log\sin\text{x}]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=(\sin\text{x})^{\cos\text{x}}[\cos\text{x}\cot\text{x}-\sin\text{x}\log\sin\text{x}]$

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 "5 Marks Questions" 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

Show that the right circular cone of least curved surface and given volume has an altitude equal to $\sqrt 2 $ times the radius of the base.

Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=\text{x}^{\text{x}}+(\sin\text{x})^\text{x}$

If $\text{x}=3\sin\text{t}-\sin3\text{t},$ $\text{y}=3\cos-\cos3\text{t}$ find $\frac{\text{dy}}{\text{dx}}$ at $\text{t}=\frac{\pi}{3}.$

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 &4 \\ 3 & 1 & 1 \end{bmatrix}$

Represent the following families of curves by forming the corresponding differential equation:
$\text{x}^2+\text{y}^2=\text{ax}^3$

Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award ₹ x each, ₹ y each and ₹ z each for the three respective values to 3, 2 and 1 students respectively with a total award money of ₹ 2,200. School Q wants to spend ₹ 3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school P). If the total amount of award for one prize on each values is ₹ 1,200, using matrices, find the award money for each value.
Apart from these three values, suggest one more value which should be considered for award.

Show that the relation R on the set Z of integers, given by R = {(a, b): 2 divides a - b}, is an equivalence relation.

A diet of two foods $F_1$ and $F_2$ contains nutrients thiamine, phosphorous and iron.
The amount of each nutrient in each of the food $($in milligrams per $25$gms$) $is given in the following table:

Nutrients	Food	$F_1$	$F_2$
Thiamine		$0.25$	$0.10$
Phosphorous		$0.75$	$1.50$
Iron		$1.60$	$0.80$

The minimum requirement of the nutrients in the diet are $1.00$mg of thiamine, $7.50$mg of phosphorous and $10.00$mg of iron.
The cost of $F_1$ is $20$ paise per $25$gms while the cost of $F_2$ is 15 paise per 25gms.
Find the minimum cost of diet.

Show that the right circular cylinder, open at the top, and of given surface area and maximum volume is such that its height is equal to the radius of the base.

Five cards are drawn successively with replacement from a well-shuffled deck of 52 cards. What is the probability that:

All the five cards are spades?
Only 3 cards are spades?
None is a spade?