Advertisements
Advertisements
प्रश्न
If x = a(1 − cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{1}{a}\text { at } \theta = \frac{\pi}{2}\] ?
Advertisements
उत्तर
Here,
\[x = a\left( 1 - \cos\theta \right) \text { and y } = a\left( \theta + \sin\theta \right)\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d x}{d \theta} = a \sin\theta \text { and }\frac{d y}{d \theta} = a + a \cos\theta\]
\[ \therefore \frac{d y}{d x} = \frac{a + a\cos\theta}{a\sin\theta} = \frac{1 + \cos\theta}{\sin\theta}\]
\[\text {Differentiating w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = \left\{ \frac{- \sin^2 \theta - \left( 1 + \cos\theta \right)\cos\theta}{\sin^2 \theta} \right\}\frac{d \theta}{d x}\]
\[ = \left( - 1 - \frac{\left( 1 + cos\theta \right)cos\theta}{\sin^2 \theta} \right)\frac{1}{a \sin\theta}\]
\[ \therefore \frac{d^2 y}{d x^2} at \theta = \frac{\pi}{2} \]
\[ \Rightarrow \left[ \frac{d^2 y}{d x^2} \right]_\theta = \frac{\pi}{2} = \frac{1}{a}\left( - 1 - \frac{0}{1} \right) = \frac{- 1}{a}\]
Hence proved.
APPEARS IN
संबंधित प्रश्न
Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[e^{\sin^{- 1} 2x}\] ?
Differentiate \[\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
If \[y = e^x + e^{- x}\] prove that \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?
Differentiate
\[\tan^{- 1} \left( \frac{\cos x + \sin x}{\cos x - \sin x} \right), \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
If \[y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty\] prove that \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
If \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
Differentiate x2 with respect to x3
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 \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right), \text { if }- \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .
Find the second order derivatives of the following function x3 + tan x ?
Find the second order derivatives of the following function tan−1 x ?
Find the second order derivatives of the following function log (log x) ?
If y = (sin−1 x)2, prove that (1 − x2)
\[\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
If y = 500 e7x + 600 e−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
If x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
If logy = tan–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if x ∈ (–1, 1) .
Find the minimum value of (ax + by), where xy = c2.
Differentiate sin(log sin x) ?
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
