Advertisements
Advertisements
प्रश्न
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
विकल्प
a constant
a function of x only
a function of y only
a function of x and y
उत्तर
(a) a constant
Here,
\[y^2 = a x^2 + bx + c\]
\[\text { Now,} \]
\[2y\frac{d y}{d x} = 2ax + b\]
\[ \Rightarrow 2y\frac{d^2 y}{d x^2} + 2 \left( \frac{d y}{d x} \right)^2 = 2a \]
\[ \Rightarrow y\frac{d^2 y}{d x^2} + \left( \frac{d y}{d x} \right)^2 = a \]
\[ \Rightarrow y\frac{d^2 y}{d x^2} + \left( \frac{2ax + b}{2y} \right)^2 = a \left[ \because 2y\frac{d y}{d x} = 2ax + b \right]\]
\[ \Rightarrow 4 y^3 \frac{d^2 y}{d x^2} + \left( 2ax + b \right)^2 = 4a y^2 \]
\[ \Rightarrow y^3 \frac{d^2 y}{d x^2} = \frac{4a y^2 - \left( 2ax + b \right)^2}{4}\]
\[ \Rightarrow y^3 \frac{d^2 y}{d x^2} = \frac{4a\left( a x^2 + bx + c \right) - \left( 2ax + b \right)^2}{4} \left[ \because y^2 = a x^2 + bx + c \right]\]
\[ \Rightarrow y^3 \frac{d^2 y}{d x^2} = \frac{4 a^2 x^2 + 4abx + 4ac - 4 a^2 x^2 - b^2 - 4axb}{4}\]
\[ \Rightarrow y^3 \frac{d^2 y}{d x^2} = \frac{4ac - b^2}{4} = \text { a constant }\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles x2ex ?
Differentiate `2^(x^3)` ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( 1 - 2 x^2 \right), 0 < x < 1\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
If \[y = x \sin \left( a + y \right)\] ,Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[x^{\cos^{- 1} x}\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
If \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
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 \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?
\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?
If \[x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,\] prove that \[\frac{dy}{dx} = - \frac{y \log x}{x \log y}\] ?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( \frac{1}{2 \sqrt{2}}, \frac{1}{2} \right)\] ?
If \[f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)\] then find \[\frac{dy}{dx} \text { at } x = 1\] ?
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 \[f\left( 0 \right) = f\left( 1 \right) = 0, f'\left( 1 \right) = 2 \text { and y } = f \left( e^x \right) e^{f \left( x \right)}\] write the value of \[\frac{dy}{dx} \text{ at x } = 0\] ?
If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
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 \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\] is equal to __________ .
If \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .
Find the second order derivatives of the following function e6x cos 3x ?
If x = a (1 − cos3 θ), y = a sin3 θ, prove that \[\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}\] ?
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
