Advertisements
Advertisements
प्रश्न
Find \[\frac{dy}{dx}\] in the following case \[\left( x^2 + y^2 \right)^2 = xy\] ?
Advertisements
उत्तर
\[\text{We have, } \left( x^2 + y^2 \right) = xy\]
Differentiating with respect to x, we get,
\[\Rightarrow \frac{d}{dx}\left[ \left( x^2 + y^2 \right)^2 \right] = \frac{d}{dx}\left( xy \right)\]
\[ \Rightarrow 2\left( x^2 + y^2 \right)\frac{d}{dx}\left( x^2 + y^2 \right) = x\frac{d y}{d x} + y\frac{d}{dx}\left( x \right) \]
\[ \Rightarrow 2\left( x^2 + y^2 \right)\left( 2x + 2y\frac{d y}{d x} \right) = x\frac{d y}{d x} + y\left( 1 \right)\]
\[ \Rightarrow 4x\left( x^2 + y^2 \right) + 4y\left( x^2 + y^2 \right)\frac{d y}{d x} = x\frac{d y}{d x} + y\]
\[ \Rightarrow 4y\left( x^2 + y^2 \right)\frac{d y}{d x} - x\frac{d y}{d x} = y - 4x\left( x^2 + y^2 \right)\]
\[ \Rightarrow \frac{d y}{d x}\left[ 4y\left( x^2 + y^2 \right) - x \right] = y - 4x\left( x^2 + y^2 \right)\]
\[ \Rightarrow \frac{d y}{d x} = \frac{y - 4x\left( x^2 + y^2 \right)}{4y\left( x^2 + y^2 \right) - x}\]
\[ \Rightarrow \frac{d y}{d x} = \frac{4x\left( x^2 + y^2 \right) - y}{x - 4y\left( x^2 + y^2 \right)}\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles log cosec x ?
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Differentiate sin (3x + 5) ?
Differentiate logx 3 ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
If \[y = x \sin^{- 1} x + \sqrt{1 - x^2}\] ,prove that \[\frac{dy}{dx} = \sin^{- 1} x\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
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 `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?
If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?
If \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Differentiate x2 with respect to x3
Differentiate (log x)x with respect to log x ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
If \[f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e\] ?
If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?
If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
If \[f\left( x \right) = \log \left\{ \frac{u \left( x \right)}{v \left( x \right)} \right\}, u \left( 1 \right) = v \left( 1 \right) \text{ and }u' \left( 1 \right) = v' \left( 1 \right) = 2\] , then find the value of `f' (1)` ?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
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 y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 0 ?
If \[y = e^{2x} \left( ax + b \right)\] show that \[y_2 - 4 y_1 + 4y = 0\] ?
If x = 2 cos t − cos 2t, y = 2 sin t − sin 2t, find \[\frac{d^2 y}{d x^2}\text{ at } t = \frac{\pi}{2}\] ?
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
