Differentiate the following functions with respect to x: x^ ^ -1 … - Vidyadip

Question

Differentiate the following functions with respect to x:
$\text{x}^{\sin^{-1}\text{x}}$

✓

Answer

Let $\text{y}=\text{x}^{\sin^{-1}\text{x}}\ .....(\text{i})$
Taking log on both the sides,
$\log\text{y}=\log\text{x}^{\sin^{-1}\text{x}}$
$\log\text{y}=\sin^{-1}\text{x}\log\text{x}$
Differentiating it with respect to x,
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\sin^{-1}\text{x}\frac{\text{d}}{\text{dx}}(\log\text{x})+(\log\text{x})\frac{\text{d}}{\text{dx}}(\sin^{-1}\text{x})$
$\Rightarrow\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\sin^{-1}\text{x}\Big(\frac{1}{\text{x}}\Big)+(\log\text{x})\Big(\frac{1}{\sqrt{1-\text{x}^2}}\Big)$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{\sin^{-1}\text{x}}{\text{x}}+\frac{\log\text{x}}{\sqrt{1-\text{x}^2}}\Big]$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\text{x}^{\sin^{-1}\text{x}}\Big[\frac{\sin^{-1}\text{x}}{\text{x}}+\frac{\log\text{x}}{\sqrt{1-\text{x}^2}}\Big]$
[Using equation (i)]

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\limits^{\frac{3}{2}}_0\big|\text{x}\cos\pi\text{x}\big|\text{dx}$

Find the value of p, so that the lines $l_1$ :$\frac{1-\text{x}}{3}=\frac{7\text{y}-\text{14}}{\text{p}}=\frac{\text{z}-\text{3}}{2}$ and $l_2$: $\frac{7-\text{7x}}{3\text{p}}=\frac{\text{y}-\text{5}}{1}=\frac{6-\text{z}}{5}$are perpendicular to each other. Also find the equations of a line passing through a point (3, 2,– 4) and parallel to line $l_1$.

Find the values of p and q so that $\text{f(x)}=\begin{cases}\text{x}^2+3\text{x}+\text{p},&\text{if x}\leq1\\\text{qx}+2,&\text{if x}>1\end{cases}$ is differentiable at x = 1.

Find the perpendicular distence of the point (1, 0, 0) from the line $\frac{\text{x}-1}{2}=\frac{\text{y}+1}{-3}=\frac{\text{z}+10}{8}.$ Also, find the coordinates of the perpendicular and the equation of the perpendicular.

Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = a - b for all a, b ∈ Q.

Prove that: $\tan^{-1}\frac{2\text{a}\text{b}}{\text{a}^2-\text{b}^2}+\tan^{-1}\frac{2\text{xy}}{\text{x}^2-\text{y}^2}=\tan^{-1}\frac{2\alpha\beta}{\alpha^2-\beta^2},$where $\alpha=\text{ax}-\text{by}$ and $\beta=\text{ay}+\text{bx}.$

Let A be the set of all human beings in a town at a particular time. Determine whether the following relations are reflexive, symmetric and transitive:
R = {(x, y): x and y work at the same place}

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}+\text{y}\cos\text{x}=\sin\text{x}\cos\text{x}$

Write the minimum value of $f(x) = x^x .$

Find the length and the foot of the perpendicular drawn from the point $(2, –1, 5)$ to the line $\frac{\text{x - 11}}{10}=\frac{\text{y + 2}}{-4}=\frac{\text{z + 8}}{-11}.$