Advertisements
Advertisements
प्रश्न
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
Advertisements
उत्तर
Here,
\[x = a\left( 1 + \cos\theta \right) \text{ and } y = a\left( \theta + \sin\theta \right)\]
\[\text{ Differentiating w . r . t .} \theta, \text{ we get }\]
\[\frac{d x}{d \theta} = - a\sin\theta \text{ and } \frac{d y}{d \theta} = a + a \cos\theta\]
\[ \therefore \frac{d y}{d x} = \frac{a + a\cos\theta}{- a\sin\theta} = \frac{1 + \cos\theta}{- \sin\theta}\]
\[\text{ Differentiating w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = \frac{d}{d\theta}\left\{ \frac{d y}{d x} \right\}\frac{d\theta}{dx}\]
\[\frac{d^2 y}{d x^2} = - \left\{ \frac{- \sin^2 \theta - \cos\theta - \cos^2 \theta}{\sin^2 \theta} \right\}\frac{d\theta}{dx}\]
\[ = \frac{1 + \cos\theta}{\sin^2 \theta} \times \frac{- 1}{a \sin\theta}\]
\[ = \frac{- \left( 1 + \cos\theta \right)}{a \sin^3 \theta}\]
\[\text{ At } \theta = \frac{\pi}{2}: \frac{d^2 y}{d x^2} = \frac{- \left( 1 + \cos\frac{\pi}{2} \right)}{a \left( \sin\frac{\pi}{2} \right)^3} = \frac{- 1}{a}\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
Differentiate \[\left( \log x \right)^{ \log x }\] ?
If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?
If \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?
Find \[\frac{dy}{dx}\] , when \[x = b \sin^2 \theta \text{ and } y = a \cos^2 \theta\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - 1}{x + 1} \right)\] with respect to \[\sin^{- 1} \left( 3x - 4 x^3 \right), \text { if }- \frac{1}{2} < x < \frac{1}{2}\] ?
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
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 \[y = \tan^{- 1} \left( \frac{1 - x}{1 + x} \right), \text{ find} \frac{dy}{dx}\] ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
If \[x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =\] ____________ .
Find the second order derivatives of the following function sin (log x) ?
Find the second order derivatives of the following function x3 log x ?
If x = sin t, y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
\[\text { If }y = A e^{- kt} \cos\left( pt + c \right), \text { prove that } \frac{d^2 y}{d t^2} + 2k\frac{d y}{d t} + n^2 y = 0, \text { where } n^2 = p^2 + k^2 \] ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
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 = etan x, then (cos2 x)y2 =
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
f(x) = xx has a stationary point at ______.
