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 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

Evaluate the following integrals:
$\int\text{cosec}^43\text{x}\text{ dx}$

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.
$\text{f(x)}=[\text{x}]\text{ for x}\in[-2,\ 2]$

A coaching institute of English (subject) conducts classes in two batches I and II and fees for rich and poor children are different. In batch I, it has 20 poor and 5 rich children and total monthly collection is ₹ 9,000, whereas in batch II, it has 5 poor and 25 rich children and total monthly collection is ₹ 26,000. Using matrix method, find monthly fees paid by each child of two types. What values the coaching institute is inculcating in the society?

Solve the following differential equation:
$\text{xy}\frac{\text{dy}}{\text{dx}}=\text{x}^2-\text{y}^2$

Find the equation of the normal to the curve $x^2 + 2y^2 - 4x - 6y + 8 = 0$ at the point whose abscissa is $2.$

There are three categories of students in a class of 60 students:

A : Very hardworking

B : Regular but not so hardworking

C : Careless and irregular 10 students are in category A, 30 in category B and the rest in category C.

It is found that the probability of students of category A, unable to get good marks in the final year examination is 0.002, of category B it is 0.02 and of category C, this probability is 0.20. A student selected at random was found to be one who could not get good marks in the examination. Find the probability that this student is category C.

Two dice are thrown together and the total score is noted. The events E, F and G are ‘a total of 4’, ‘a total of 9 or more’, and ‘a total divisible by 5’, respectively. Calculate P(E), P(F) and P(G) and decide which pairs of events, if any, are independent.

Find the distance of the point (-1, -5, -10) from the point of intersection of the line $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}}+\lambda\Big(3\hat{\text{i}}+4\hat{\text{j}}+2\hat{\text{k}}\Big)$ and the plane $\vec{\text{r}}.\Big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\Big)=5.$

Evaluvate the following intregals:
$\int\frac{1}{3+4\cot\text{x}}\ \text{dx}$

A company has two plants to manufacture bicycles. The first plant manufactures $60\%$ of the bicycles and the second plant $40\%.$ Out of the $80\%$ of the bicycles are rated of standard quality at the first plant and $90\%$ of standard quality at the second plant. $A$ bicycle is picked up at random and found to be standard quality. Find the probability that it comes from the second plant.