Advertisements
Advertisements
प्रश्न
Find \[\frac{dy}{dx}\] in the following case: \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?
उत्तर
\[\text{ We have }, y^3 - 3x y^2 = x^3 + 3 x^2 y\]
Differentiating with respect to x, we get,
\[\Rightarrow \frac{d}{dx}\left( y^3 \right) - \frac{d}{dx}\left( 3x y^2 \right) = \frac{d}{dx}\left( x^3 \right) + \frac{d}{dx}\left( 3 x^2 y \right)\]
\[ \Rightarrow 3 y^2 \frac{d y}{d x} - 3\left[ x\frac{d}{dx}\left( y^2 \right) + y^2 \frac{d}{dx}\left( x \right) \right] = 3 x^2 + 3\left[ x^2 \frac{d}{dx}\left( y \right) + y\frac{d}{dx}\left( x^2 \right) \right] \left[ \text{ Using product rule } \right]\]
\[ \Rightarrow 3 y^2 \frac{d y}{d x} - 3\left[ x\left( 2y \right)\frac{d y}{d x} + y^2 \right] = 3 x^2 + 3\left[ x^2 \frac{d y}{d x} + y\left( 2x \right) \right]\]
\[ \Rightarrow 3 y^2 \frac{d y}{d x} - 6xy\frac{d y}{d x} - 3 y^2 = 3 x^2 + 3 x^2 \frac{d y}{d x} + 6xy\]
\[ \Rightarrow 3 y^2 \frac{d y}{d x} - 6xy\frac{d y}{d x} - 3 x^2 \frac{d y}{d x} = 3 x^2 + 6xy + 3 y^2 \]
\[ \Rightarrow 3\frac{d y}{d x}\left( y^2 - 2xy - x^2 \right) = 3\left( x^2 + 2xy + y^2 \right)\]
\[ \Rightarrow \frac{d y}{d x} = \frac{3 \left( x + y \right)^2}{3\left( y^2 - 2xy - x^2 \right)}\]
\[ \Rightarrow \frac{d y}{d x} = \frac{\left( x + y \right)^2}{y^2 - 2xy - x^2}\]
APPEARS IN
संबंधित प्रश्न
Differentiate sin (log x) ?
Differentiate sin2 (2x + 1) ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\] ?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
Differentiate \[\log \left( \cos x^2 \right)\] ?
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\], prove that \[2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\] ?
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 \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\cot x} + \left( \cot x \right)^{\tan x}\] ?
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
If \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
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 f (x) is an even function, then write whether `f' (x)` is even or odd ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
Let \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\] ____________ .
If x = a (θ + sin θ), y = a (1 + cos θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{a}{y^2}\] ?
If `x = sin(1/2 log y)` show that (1 − x2)y2 − xy1 − a2y = 0.
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
If x = t2, y = t3, then \[\frac{d^2 y}{d x^2} =\]
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
