Advertisements
Advertisements
प्रश्न
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
उत्तर
\[\text { Let, u }= \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\]
\[ \Rightarrow u = \tan^{- 1} \left[ \tan\left( \frac{\pi}{4} - \frac{x}{2} \right) \right]\]
\[ \Rightarrow u = \frac{\pi}{4} - \frac{x}{2}\]
Differentiating it with respect to x,
\[\frac{du}{dx} = 0 - \left( \frac{1}{2} \right)\]
\[\frac{du}{dx} = - \frac{1}{2} . . . \left( i \right)\]
\[\text { Let, v } = se c^{- 1} x\]
Differentiating it with respect to x,
\[\frac{dv}{dx} = \frac{1}{x\sqrt{x^2 - 1}} . . . \left( ii \right)\]
\[\text { Dividing equation } \left( i \right) \text { by}\left( ii \right), \]
\[\frac{\frac{du}{dx}}{\frac{dv}{dx}} = - \frac{1}{2} \times \frac{x\sqrt{x^2 - 1}}{1}\]
\[\frac{du}{dv} = \frac{- x\sqrt{x^2 - 1}}{2}\]
APPEARS IN
संबंधित प्रश्न
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`
Differentiate the following functions from first principles eax+b.
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
If \[y = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] .prove that \[\frac{dy}{dx} = 1 - y^2\] ?
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[\left( x^x \right) \sqrt{x}\] ?
Differentiate \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?
If \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
Find the derivative of the function f (x) given by \[f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)\] and hence find `f' (1)` ?
If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
Differentiate x2 with respect to x3
Differentiate (log x)x with respect to log x ?
If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .
If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{ then} \frac{dy}{dx} =\] ____________ .
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
Find the second order derivatives of the following function log (sin x) ?
If x = a(1 − cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}\] ?
Find \[\frac{d^2 y}{d x^2}\] where \[y = \log \left( \frac{x^2}{e^2} \right)\] ?
If \[y = e^{a \cos^{- 1}} x\] ,prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} - a^2 y = 0\] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
If x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
Differentiate `log [x+2+sqrt(x^2+4x+1)]`
