Question
Differentiate the following functions with respect to x:
$\sin(2\sin^{-1}\text{x})$
$\sin(2\sin^{-1}\text{x})$
✓
Answer
Let, $\text{y}=\sin(2\sin^{-1}\text{x})$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big(\sin(2\sin^{-1}\text{x})\Big)$
$=\cos\big(2\sin^{-1}\text{x}\big)\frac{\text{d}}{\text{dx}}\big(2\sin^{-1}\text{x}\big)$
[Using chain rule]
$=\cos\big(2\sin^{-1}\text{x}\big)\times2\frac{1}{\sqrt{1-\text{x}^2}}$
$=\frac{2\cos\big(2\sin^{-1}\text{x}\big)}{\sqrt{1-\text{x}^2}}$
So,
$\frac{\text{d}}{\text{dx}}\Big(\sin\big(2\sin^{-1}\text{x}\big)\Big)=\frac{2\cos\big(2\sin^{-1}\text{x}\big)}{\sqrt{1-\text{x}^2}}$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big(\sin(2\sin^{-1}\text{x})\Big)$
$=\cos\big(2\sin^{-1}\text{x}\big)\frac{\text{d}}{\text{dx}}\big(2\sin^{-1}\text{x}\big)$
[Using chain rule]
$=\cos\big(2\sin^{-1}\text{x}\big)\times2\frac{1}{\sqrt{1-\text{x}^2}}$
$=\frac{2\cos\big(2\sin^{-1}\text{x}\big)}{\sqrt{1-\text{x}^2}}$
So,
$\frac{\text{d}}{\text{dx}}\Big(\sin\big(2\sin^{-1}\text{x}\big)\Big)=\frac{2\cos\big(2\sin^{-1}\text{x}\big)}{\sqrt{1-\text{x}^2}}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.
Without using the concept of inverse of a matrix, find the matrix $\begin{bmatrix}\text{x}&\text{y}\\\text{z}&\text{u}\end{bmatrix}$ such that $\begin{bmatrix}5&-7\\-2&3\end{bmatrix}\begin{bmatrix}\text{x}&\text{y}\\\text{z}&\text{u}\end{bmatrix}=\begin{bmatrix}-16&-6\\7&2\end{bmatrix}$
→Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\text{a}\sin\text{x}+\text{b}\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$
→$\int\limits^{\frac{\pi}{2}}_0\frac{\text{a}\sin\text{x}+\text{b}\sin\text{x}}{\sin\text{x}+\cos\text{x}}\text{ dx}$
Find the direction cosines of the lines, connected by the relations: $l + m + n = 0$ and $\frac{2}{\text{m}}+\frac{2}{\text{n}}-\text{mn}=0$.
→A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of 
17.50 per package on nuts and 
7 per package of bolts. How many packages of each should be produced each day so as to maximise his profits if he operates his machines for at the most 12 hours a day? Form the above as a linear programming problem and solve it graphically.
17.50 per package on nuts and 7 per package of bolts. How many packages of each should be produced each day so as to maximise his profits if he operates his machines for at the most 12 hours a day? Form the above as a linear programming problem and solve it graphically.Find A, if $\begin{bmatrix}4\\1\\3\end{bmatrix}\text{A}=\begin{bmatrix}-4&8&4\\-1&2&1\\-3&6&3\end{bmatrix}.$
→Differentiate the following functions with respect to x:
$(\cos\text{x})^\text{x}+(\sin\text{x})^\frac{1}{\text{x}}$
→$(\cos\text{x})^\text{x}+(\sin\text{x})^\frac{1}{\text{x}}$
A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent daily on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimize cost.
Prove that the lines through A(0, -1, -1) and B(4, 5, 1) intersects the line through C(3, 9, 4) and D(-4, 4, 4). Also, find their point of intersection.
A bag contains $4$ white and $5$ black balls. Another bag contains $9$ white and $7$ black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.
→