Advertisements
Advertisements
प्रश्न
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]
विकल्प
f(t) − f''(t)
{f(t) − f'' (t)}2
{f(t) + f''(t)}2
none of these
Advertisements
उत्तर
(c){f(t) + f''(t)}2
Here,
\[x = f\left( t \right)\cos t - f^{'} \left( t \right) \sin t \text { and y } = f\left( t \right) \sin t + f^{'} \left( t \right)\cos t\]
\[ \Rightarrow \frac{d x}{d t} = f^{'} \left( t \right)\cos t - f\left( t \right)\sin t - f^{''} \left( t \right)\sin t - f^{'} \left( t \right)\cos t \text { and } \frac{d y}{d t} = f^{'} \left( t \right) \sin t + f\left( t \right)\cos t + f^{''} \left( t \right)\cos t - f^{'} \left( t \right) \sin t\]
\[ \Rightarrow \frac{d x}{d t} = - f\left( t \right)\sin t - f^{''} \left( t \right)\sin t \text { and } \frac{d y}{d t} = f\left( t \right)\cos t + f^{''} \left( t \right)\cos t\]
\[\text { Thus }, \]
\[ \left( \frac{d x}{d t} \right)^2 + \left( \frac{d y}{d t} \right)^2 = \left\{ - f\left( t \right)\sin t - f^{''} \left( t \right)\sin t \right\}^2 + \left\{ f\left( t \right)\cos t + f^{''} \left( t \right)\cos t \right\}^2 \]
\[ = \left\{ f\left( t \right)\sin t + f^{''} \left( t \right)\sin t \right\}^2 + \left\{ f\left( t \right)\cos t + f^{''} \left( t \right)\cos t \right\}^2 \]
\[ = \sin^2 t \left\{ f\left( t \right) + f^{''} \left( t \right) \right\}^2 + \cos^2 t \left\{ f\left( t \right) + f^{''} \left( t \right) \right\}^2 \]
\[ = \left\{ f\left( t \right) + f^{''} \left( t \right) \right\}^2 \left( \sin^2 t + \cos^2 t \right)\]
\[ = \left\{ f\left( t \right) + f^{''} \left( t \right) \right\}^2\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles log cosec x ?
Differentiate tan (x° + 45°) ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?
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 \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
Find \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?
If \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[e^{x \log x}\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
Differentiate x2 with respect to x3
If \[f\left( 1 \right) = 4, f'\left( 1 \right) = 2\] find the value of the derivative of \[\log \left( f\left( e^x \right) \right)\] w.r. to x at the point x = 0 ?
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 \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}\] ?
Find the second order derivatives of the following function x3 log x ?
If y = e−x cos x, show that \[\frac{d^2 y}{d x^2} = 2 e^{- x} \sin x\] ?
If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If y = 3 cos (log x) + 4 sin (log x), prove that x2y2 + xy1 + y = 0 ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
If x = 2 at, y = at2, where a is a constant, then \[\frac{d^2 y}{d x^2} \text { at x } = \frac{1}{2}\] is
Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
