Advertisements
Advertisements
प्रश्न
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
विकल्प
3(xy2 + y1)y2
3(xy1 + y2)y2
3(xy2 + y1)y1
none of these
उत्तर
(a) 3(xy2 + y1)y2
Here,
\[y = \frac{ax + b}{x^2 + c}\]
\[ \Rightarrow \left( x^2 + c \right)y = ax + b\]
\[\text { Diffferentiating w . r . t . x, we get }\]
\[2xy + \left( x^2 + c \right)\frac{dy}{dx} = a\]
\[\text { Diffferentiating w . r . t . x, we get }\]
\[2y + 2x y_1 + 2x y_1 + \left( x^2 + c \right) y_2 = 0\]
\[ \Rightarrow 2y + 4x y_1 + \left( x^2 + c \right) y_2 = 0\]
\[\text { Diffferentiating again w . r . t . x, we get }\]
\[2 y_1 + 4 y_1 + 4x y_2 + \left( x^2 + c \right) y_3 + 2x y_2 = 0\]
\[ \Rightarrow 6 y_1 + 6x y_2 + \left( x^2 + c \right) y_3 = 0\]
\[ \Rightarrow 6 y_1 + 6x y_2 + \left( \frac{- 2y - 4x y_1}{y_2} \right) y_3 = 0 \left[ \because 2y + 4x y_1 + \left( x^2 + c \right) y_2 = 0 \right]\]
\[ \Rightarrow 6 y_1 y_2 + 6x \left( y_2 \right)^2 - 2y - 4x y_1 y_3 = 0\]
\[ \Rightarrow 3 y_1 y_2 + 3x \left( y_2 \right)^2 - y - 2x y_1 y_3 = 0\]
\[ \Rightarrow \left( y_1 + x y_2 \right)3 y_2 = \left( 2x y_1 + y \right) y_3 \]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles x2ex ?
Differentiate logx 3 ?
Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{\sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{2 a^x}{1 - a^{2x}} \right), a > 1, - \infty < x < 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
If \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\], show that \[\frac{dy}{dx}\] is independent of x. ?
If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
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 \[e^{x \log x}\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Differentiate \[x^{\tan^{- 1} x }\] ?
Find \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
Find \[\frac{dy}{dx}\] ,when \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?
If \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
If f (x) = loge (loge x), then write the value of `f' (e)` ?
If \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
If \[x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =\] ____________ .
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
Find the second order derivatives of the following function sin (log x) ?
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
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 = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
\[\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3} \] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
\[\text { If y } = a \left\{ x + \sqrt{x^2 + 1} \right\}^n + b \left\{ x - \sqrt{x^2 + 1} \right\}^{- n} , \text { prove that }\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \]
Disclaimer: There is a misprint in the question,
\[\left( x^2 + 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0\] must be written instead of
\[\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + x\frac{d y}{d x} - n^2 y = 0 \] ?
If y = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
