Advertisements
Advertisements
प्रश्न
If \[x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}\] ?
उत्तर
\[\text{ We have, x} = a\left( \theta + \sin\theta \right) \text{ and y }= a\left( 1 + \cos\theta \right)\]
\[ \Rightarrow \frac{dx}{d\theta} = a\left\{ \frac{d}{d\theta}\left( \theta \right) + \frac{d}{d\theta}\left( \sin\theta \right) \right\} \text{ and }\frac{dy}{d\theta} = a\left( 0 - \sin\theta \right)\]
\[ \Rightarrow \frac{dx}{d\theta} = a\left( 1 + \cos\theta \right) \text{ and } \frac{dy}{d\theta} = - a\sin\theta \]
\[ \therefore \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{- a\sin\theta}{a\left( 1 + \cos\theta \right)} = \frac{- 2\sin\frac{\theta}{2}\cos\frac{\theta}{2}}{2 \cos^2 \frac{\theta}{2}} = - \tan\frac{\theta}{2} \]
APPEARS IN
संबंधित प्रश्न
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate (log sin x)2 ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
If \[\log \sqrt{x^2 + y^2} = \tan^{- 1} \left( \frac{y}{x} \right)\] Prove that \[\frac{dy}{dx} = \frac{x + y}{x - y}\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[\left( \sin x \right)^{\log x}\] ?
Differentiate \[x^{\tan^{- 1} x }\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
Find \[\frac{dy}{dx}\] \[y = \sin x \sin 2x \sin 3x \sin 4x\] ?
If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?
If \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
Find \[\frac{dy}{dx}\], when \[x = a t^2 \text{ and } y = 2\ at \] ?
If \[\frac{dy}{dx}\] when \[x = a \cos \theta \text{ and } y = b \sin \theta\] ?
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 \[\tan^{- 1} \left( \frac{1 + ax}{1 - ax} \right)\] with respect to \[\sqrt{1 + a^2 x^2}\] ?
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
If \[f\left( x \right) = \sqrt{x^2 + 6x + 9}, \text { then } f'\left( x \right)\] is equal to ______________ .
If \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\] is equal to __________ .
Find the second order derivatives of the following function sin (log x) ?
Find the second order derivatives of the following function x cos x ?
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 ?
\[\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text { and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2} \] ?
If y = a xn + 1 + bx−n and \[x^2 \frac{d^2 y}{d x^2} = \lambda y\] then write the value of λ ?
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
If logy = tan–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`
\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]
\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]
Find the minimum value of (ax + by), where xy = c2.
