Advertisements
Advertisements
प्रश्न
If \[x = a \left( \frac{1 + t^2}{1 - t^2} \right) \text { and y } = \frac{2t}{1 - t^2}, \text { find } \frac{dy}{dx}\] ?
उत्तर
\[ \Rightarrow \frac{dx}{dt} = a\left[ \frac{\left( 1 - t^2 \right)\left( 2t \right) - \left( 1 + t^2 \right)\left( - 2t \right)}{\left( 1 - t^2 \right)^2} \right]\]
\[ \Rightarrow \frac{dx}{dt} = a\left[ \frac{2t - 2 t^3 + 2t + 2 t^3}{\left( 1 - t^2 \right)^2} \right]\]
\[ \Rightarrow \frac{dx}{dt} = \frac{4at}{\left( 1 - t^2 \right)^2} . . . \left( i \right)\]
\[\text { and,} \]
\[ y = \frac{2t}{1 - t^2}\]
\[ \Rightarrow \frac{dy}{dt} = 2\left[ \frac{\left( 1 - t^2 \right)\left( 1 \right) - t\left( - 2t \right)}{\left( 1 - t^2 \right)^2} \right]\]
\[ \Rightarrow \frac{dy}{dt} = 2\left[ \frac{1 - t^2 + 2 t^2}{\left( 1 - t^2 \right)^2} \right]\]
\[ \Rightarrow \frac{dy}{dt} = \frac{2\left( 1 + t^2 \right)}{\left( 1 - t^2 \right)^2} . . . \left( ii \right)\]
\[\text { Dividing equation } \left( ii \right) \text { by } \left( i \right), \]
\[\frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{2\left( 1 + t^2 \right)}{\left( 1 - t^2 \right)^2} \times \frac{\left( 1 - t^2 \right)^2}{4at}\]
\[ \Rightarrow \frac{dy}{dx} = \frac{\left( 1 + t^2 \right)}{2at}\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles e−x.
Differentiate \[3^{e^x}\] ?
Differentiate logx 3 ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
Differentiate \[e^x \log \sin 2x\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
If \[y = x \sin^{- 1} x + \sqrt{1 - x^2}\] ,prove that \[\frac{dy}{dx} = \sin^{- 1} x\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
If \[y = \cos^{- 1} \left( 2x \right) + 2 \cos^{- 1} \sqrt{1 - 4 x^2}, - \frac{1}{2} < x < 0, \text{ find } \frac{dy}{dx} \] ?
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
If \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[{10}^\left( {10}^x \right)\] ?
Differentiate \[\left( x \cos x \right)^x + \left( x \sin x \right)^{1/x}\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
If \[y = \log\frac{x^2 + x + 1}{x^2 - x + 1} + \frac{2}{\sqrt{3}} \tan^{- 1} \left( \frac{\sqrt{3} x}{1 - x^2} \right), \text{ find } \frac{dy}{dx} .\] ?
If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?
If \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} \text{ at }\theta = \frac{\pi}{3} \] ?
If \[x = a\sin2t\left( 1 + \cos2t \right) \text { and y } = b\cos2t\left( 1 - \cos2t \right)\] , show that at \[t = \frac{\pi}{4}, \frac{dy}{dx} = \frac{b}{a}\] ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\] ?
If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
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 \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\] is equal to __________ .
If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .
Find the second order derivatives of the following function x3 + tan x ?
Find the second order derivatives of the following function x cos x ?
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
If y = x3 log x, prove that \[\frac{d^4 y}{d x^4} = \frac{6}{x}\] ?
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 \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
Differentiate sin(log sin x) ?
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
