Advertisements
Advertisements
Question
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
Solution
\[\text{ We have, x } = e^\theta \left( \theta + \frac{1}{\theta} \right)\]
Differentiating it with respect to \[\theta\]
\[\frac{dx}{d\theta} = e^\theta \frac{d}{d\theta}\left( \theta + \frac{1}{\theta} \right) + \left( \theta + \frac{1}{\theta} \right)\frac{d}{d\theta}\left( e^\theta \right) \left[ \text{ using product rule } \right]\]
\[ \Rightarrow \frac{dx}{d\theta} = e^\theta \left( 1 - \frac{1}{\theta^2} \right) + \left( \frac{\theta^2 + 1}{\theta} \right) e^\theta \]
\[ \Rightarrow \frac{dx}{d\theta} = e^\theta \left( 1 - \frac{1}{\theta^2} + \frac{\theta^2 + 1}{\theta} \right)\]
\[ \Rightarrow \frac{dx}{d\theta} = e^\theta \left( \frac{\theta^2 - 1 + \theta^3 + \theta}{\theta^2} \right)\]
\[ \Rightarrow \frac{dx}{d\theta} = \frac{e^\theta \left( \theta^3 + \theta^2 + \theta - 1 \right)}{\theta^2} . . . \left( i \right)\]
\[\text{ and }, \]
\[ y = e^\theta \left( \theta - \frac{1}{\theta} \right)\]
Differentiating it with respect to \[\theta\] using chain rule
\[\frac{dy}{d\theta} = e^{- \theta} \frac{d}{d\theta}\left( \theta - \frac{1}{\theta} \right) + \left( \theta - \frac{1}{\theta} \right)\frac{d}{d\theta}\left( e^{- \theta} \right) \left[ \text{ using product rule } \right]\]
\[ \Rightarrow \frac{dy}{d\theta} = e^{- \theta} \left( 1 + \frac{1}{\theta^2} \right) + \left( \theta - \frac{1}{\theta} \right) e^\theta \frac{d}{d\theta}\left( - \theta \right)\]
\[ \Rightarrow \frac{dy}{d\theta} = e^{- \theta} \left( 1 + \frac{1}{\theta^2} \right) + \left( \theta - \frac{1}{\theta} \right) e^{- \theta} \left( - 1 \right)\]
\[ \Rightarrow \frac{dy}{d\theta} = e^{- \theta} \left( 1 + \frac{1}{\theta^2} - \theta + \frac{1}{\theta} \right)\]
\[ \Rightarrow \frac{dy}{d\theta} = e^{- \theta} \left( \frac{\theta^2 + 1 - \theta^3 + \theta}{\theta^2} \right)\]
\[ \Rightarrow \frac{dy}{d\theta} = \frac{e^{- \theta} \left( - \theta^3 + \theta^2 + \theta + 1 \right)}{\theta^2} . . . \left( ii \right)\]
\[\text{ Dividing equation } \left( ii \right) by \left( i \right), \]
\[\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = e^{- \theta} \left( \frac{\theta^2 - \theta^3 + \theta + 1}{\theta^2} \right) \times \frac{\theta^2}{e^\theta \left( \theta^3 + \theta^2 + \theta - 1 \right)}\]
\[ = e^{- 2\theta} \left( \frac{\theta^2 - \theta^3 + \theta + 1}{\theta^3 + \theta^2 + \theta - 1} \right)\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles x2ex ?
Differentiate tan (x° + 45°) ?
Differentiate \[3^{e^x}\] ?
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
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 \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\] ?
Find \[\frac{dy}{dx}\] in the following case \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Find \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
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 \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?
If \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
Differential coefficient of sec(tan−1 x) is ______.
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
The derivative of \[\sec^{- 1} \left( \frac{1}{2 x^2 + 1} \right) \text { w . r . t }. \sqrt{1 + 3 x} \text { at } x = - 1/3\]
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] 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 ____________ .
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
If y = ae2x + be−x, show that, \[\frac{d^2 y}{d x^2} - \frac{dy}{dx} - 2y = 0\] ?
\[\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3} \] ?
If \[x = 3 \cos t - 2 \cos^3 t, y = 3\sin t - 2 \sin^3 t,\] find \[\frac{d^2 y}{d x^2} \] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
If y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If x = a cos nt − b sin nt, then \[\frac{d^2 x}{d t^2}\] is
If \[y^\frac{1}{n} + y^{- \frac{1}{n}} = 2x, \text { then find } \left( x^2 - 1 \right) y_2 + x y_1 =\] ?
f(x) = 3x2 + 6x + 8, x ∈ R
