Advertisements
Advertisements
प्रश्न
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Advertisements
उत्तर
\[\text{ We have, x } = a e^\theta \left( \sin\theta - \cos\theta \right) \text{ and } y = a e^\theta \left( \sin\theta + \cos\theta \right)\]
\[\Rightarrow \frac{dx}{d\theta} = a\left[ e^\theta \frac{d}{d\theta}\left( \sin\theta - \cos\theta \right) + \left( \sin\theta - \cos\theta \right)\frac{d}{d\theta}\left( e^\theta \right) \right] \text{ and } \frac{dy}{d\theta} = a\left[ e^\theta \frac{d}{d\theta}\left( \sin\theta + \cos\theta \right) + \left( \sin\theta + \cos\theta \right)\frac{d}{d\theta}\left( e^\theta \right) \right]\]
\[ \Rightarrow \frac{dx}{d\theta} = a\left[ e^\theta \left( \cos\theta + \sin\theta \right) + \left( \sin\theta - \cos\theta \right) e^\theta \right] \text{ and } \frac{dy}{d\theta} = a\left[ e^\theta \left( \cos\theta - \sin\theta \right) + \left( \sin\theta + \cos\theta \right) e^\theta \right]\]
\[ \Rightarrow \frac{dx}{d\theta} = a\left[ 2 e^\theta \sin\theta \right] \text{ and } \frac{dy}{d\theta} = a\left[ 2 e^\theta \cos\theta \right] \]
\[\therefore \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a\left( 2 e^\theta \cos\theta \right)}{a\left( 2 e^\theta \sin\theta \right)} = \cot\theta\]
APPEARS IN
संबंधित प्रश्न
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
If the function f(x)=2x3−9mx2+12m2x+1, where m>0 attains its maximum and minimum at p and q respectively such that p2=q, then find the value of m.
Differentiate sin2 (2x + 1) ?
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[\sin \left( 2 \sin^{- 1} x \right)\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
If \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?
If \[y = e^x + e^{- x}\] prove that \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Differentiate \[\left( \cos x \right)^{\sin x }\] with respect to \[\left( \sin x \right)^{\cos x }\]?
\[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?
If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
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 \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
If y = ae2x + be−x, show that, \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\] ?
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 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
\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]
\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
