Advertisements
Advertisements
प्रश्न
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
उत्तर
It is given that , `x=a (cos t+t sin t)and y=a(sin t-t cos t)`
`∴ dx/dt=a.d/dt (cos t+t sin t)`
`= a [-sin t + sin t. d/dt(t)+t.d/dt(sin t)]`
=`a[-sin t+sin t+t cos t]=at cos t`
`dy/dt=a. d/dt(sin t- cost t)`
`=a[cos t-{cos t.d/dt(t)+t. d/dt(cos t)}]`
`=a[cos t-{cos t-t sin t}]=at sin t`
`∴ dy/dx=((dy/dt))/((dx/dt))=(at sin t)/(at cos t)=tan t`
Then, ` d^2 y/dx^2=d/dx (dy/dx)=d/dx(tan t)=sec^2 t. dt/dx`
`=sec^2 t. 1/(at cos t) [dx/dt=at cost ⇒ dt/dx=1/(at cos t)]`
`= sec^3t/(at), 0<t< pi/2`
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles x2ex ?
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Differentiate etan x ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
If \[y = e^x \cos x\] ,prove that \[\frac{dy}{dx} = \sqrt{2} e^x \cdot \cos \left( x + \frac{\pi}{4} \right)\] ?
If \[y = \sqrt{a^2 - x^2}\] prove that \[y\frac{dy}{dx} + x = 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
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 \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?
If \[y = \sin \left( x^x \right)\] prove that \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?
If \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?
If \[y = \left( \cos x \right)^{\left( \cos x \right)^{\left( \cos x \right) . . . \infty}}\],prove that \[\frac{dy}{dx} = - \frac{y^2 \tan x}{\left( 1 - y \log \cos x \right)}\]?
Find \[\frac{dy}{dx}\] , when \[x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}\] ?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1\] ?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text{ find } \frac{dy}{dx}\] ?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
If \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .
Find the second order derivatives of the following function sin (log x) ?
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
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 \[y = e^{2x} \left( ax + b \right)\] show that \[y_2 - 4 y_1 + 4y = 0\] ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
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 = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
