Advertisements
Advertisements
प्रश्न
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
उत्तर
\[\text{ We have, }\sin\left( xy \right) + \frac{y}{x} = x^2 - y^2\]
Differentiating with respect to x, we get,
\[\Rightarrow \frac{d}{dx}\left( \sin xy \right) + \frac{d}{dx}\left( \frac{y}{x} \right) = \frac{d}{dx}\left( x^2 \right) - \frac{d}{dx}\left( y^2 \right)\]
\[ \Rightarrow \cos\left( xy \right)\frac{d}{dx}\left( xy \right) + \left\{ \frac{x\frac{dy}{dx} - y\frac{d}{dx}\left( x \right)}{x^2} \right\} = 2x - 2y\frac{dy}{dx} \]
\[ \Rightarrow \cos\left( xy \right)\left\{ x\frac{dy}{dx} + y\frac{d}{dx}\left( x \right) \right\} + \left\{ \frac{x\frac{dy}{dx} - y\left( 1 \right)}{x^2} \right\} = 2x - 2y\frac{dy}{dx}\]
\[ \Rightarrow \cos\left( xy \right)\left\{ x\frac{dy}{dx} + y\left( 1 \right) \right\} + \frac{1}{x^2}\left( x\frac{dy}{dx} - y \right) = 2x - 2y\frac{dy}{dx}\]
\[ \Rightarrow x \cos\left( xy \right)\frac{dy}{dx} + y \cos\left( xy \right) + \frac{1}{x}\frac{dy}{dx} - \frac{y}{x^2} = 2x - 2y\frac{dy}{dx}\]
\[ \Rightarrow \frac{dy}{dx}\left\{ x \cos\left( xy \right) + \frac{1}{x} + 2y \right\} = \frac{y}{x^2} - y \cos\left( xy \right) + 2x\]
\[ \Rightarrow \frac{dy}{dx}\left\{ \frac{x^2 \cos\left( xy \right) + 1 + 2xy}{x} \right\} = \frac{1}{x^2}\left( y - x^2 y \cos\left( xy \right) + 2 x^3 \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2 x^3 + y - x^2 y \cos\left( xy \right)}{x\left( x^2 \cos\left( xy \right) + 1 + 2xy \right)}\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
If \[y = se c^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right), x > 0 . \text{ Find} \frac{dy}{dx}\] ?
Differentiate \[x^{\tan^{- 1} x }\] ?
Differentiate \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
If \[y = \sin \left( x^x \right)\] prove that \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?
If \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Differentiate \[\sin^{- 1} \left( 2 ax \sqrt{1 - a^2 x^2} \right)\] with respect to \[\sqrt{1 - a^2 x^2}, \text{ if }-\frac{1}{\sqrt{2}} < ax < \frac{1}{\sqrt{2}}\] ?
If \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?
If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\] ______________ .
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
Find the second order derivatives of the following function e6x cos 3x ?
If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
If y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to
If \[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\] then (1 − x)2 f '' (x) − xf(x) =
If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
