Differentiate the following w.r.t.x: ( ^ -1 e^ -x ) - Vidyadip

Question

Differentiate the following w.r.t.x: $\sin(\tan^{-1}\text{e}^{-\text{x}})$

✓

Answer

$\text{Let}\ \text{y}=\sin(\tan^{-1}\text{e}^{-\text{x}})$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\cos(\tan^{-1}\text{e}^{-\text{x}}).\frac{\text{d}}{\text{dx}}(\tan^{-1}\text{e}^{-\text{x}})= \bigg[\because\frac{\text{d}}{\text{dx}}\sin\text{f(x)}=\cos\text{f(x)}\frac{\text{d}}{\text{dx}}\text{f(x)}\bigg]$
$=\cos(\tan^{-1}\text{e}^{-\text{x}}).\frac{1}{1+(\text{e}^{-\text{x}})^{2}}\frac{\text{d}}{\text{dx}}\text{e}^{-\text{x}}= \bigg[\because\frac{\text{d}}{\text{dx}}\ \tan^{-1} \text{f(x)}=\frac{1}{(\text{f(x)})^{2}}\frac{\text{d}}{\text{dx}}\text{f(x)}\bigg]$
$=\cos(\tan^{-1}\text{e}^{-\text{x}}).\frac{1}{1+\text{e}^{-\text{2x}}}\text{e}^{\text{-x}}\frac{\text{d}}{\text{dx}}({-\text{x}})$
$=\frac{-\text{e}^{-\text{x}}.\cos(\tan^{-1}\text{e}^{-\text{x}})}{1+\text{e}^{-2\text{x}}}$

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" 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

Find the maximum value of $\sin \theta+\cos \theta$.

Find the equation of the line passing through the points (1, 2, -4) and parallel to the line $\frac{\text{x}-3}{4}=\frac{\text{y}-5}{2}=\frac{\text{z}+1}{3}.$

Consider f : N → N, g : N → N and h : N → R defined as f(x) = 2x, g(y) = 3y + 4 and $\text{h(z)}=\sin\text{z}$ for all $\text{x, y, z}\in\text{N.}$ Show that ho(gof) = (hog)of.

Solve the following differential equation:

$\frac{dy}{dx} -\frac{y}{x} = 2x^{2}$

Let A = R₀ × R, where R₀ denote the set of all non-zero real numbers. A binary operation '⊙' is defined on A as follows:
(a, b) ⊙ (c, d) = (ac, bc + d) for all (a, b), (c, d) ∈ R₀ × R.
Find the invertible elements in A.

If $\text{A}=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix},$ then verify that A^TA = I₂.

Form the differential equation having $\text{y}=(\sin^{-1}\text{x})+\text{A}\cos^{-1}\text{x}+\text{B}$ where A and B are aribitrary constants, as its general solution.

Find the particular solution of the differential equation $(1+\text{e}^{2\text{x}})\text{dy}+(1+\text{y}^2)\text{e}^{\text{x}}\ \text{dx}=0,\ \text{given that y}=1\ \text{when x}=0. $

A plabne makes intercepts -6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it.

verify that $\text{y}=\text{cx}+2\text{c}^2$ is a solution of the differential equation $2\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)^2-\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$