Advertisements
Advertisements
Question
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
Solution
\[\text{ We have, x } = a e^\theta \left( \sin\theta - \cos\theta \right) \text{ and } y = a e^\theta \left( \sin\theta + \cos\theta \right)\]
\[\Rightarrow \frac{dx}{d\theta} = a\left[ e^\theta \frac{d}{d\theta}\left( \sin\theta - \cos\theta \right) + \left( \sin\theta - \cos\theta \right)\frac{d}{d\theta}\left( e^\theta \right) \right] \text{ and } \frac{dy}{d\theta} = a\left[ e^\theta \frac{d}{d\theta}\left( \sin\theta + \cos\theta \right) + \left( \sin\theta + \cos\theta \right)\frac{d}{d\theta}\left( e^\theta \right) \right]\]
\[ \Rightarrow \frac{dx}{d\theta} = a\left[ e^\theta \left( \cos\theta + \sin\theta \right) + \left( \sin\theta - \cos\theta \right) e^\theta \right] \text{ and } \frac{dy}{d\theta} = a\left[ e^\theta \left( \cos\theta - \sin\theta \right) + \left( \sin\theta + \cos\theta \right) e^\theta \right]\]
\[ \Rightarrow \frac{dx}{d\theta} = a\left[ 2 e^\theta \sin\theta \right] \text{ and } \frac{dy}{d\theta} = a\left[ 2 e^\theta \cos\theta \right] \]
\[\therefore \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a\left( 2 e^\theta \cos\theta \right)}{a\left( 2 e^\theta \sin\theta \right)} = \cot\theta\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles e3x.
Differentiate the following function from first principles \[e^\sqrt{\cot x}\] .
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate tan 5x° ?
Differentiate \[3^{e^x}\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Find \[\frac{dy}{dx}\] in the following case: \[y^3 - 3x y^2 = x^3 + 3 x^2 y\] ?
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 \[\sin \left( x^x \right)\] ?
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
Differentiate \[x^{x^2 - 3} + \left( x - 3 \right)^{x^2}\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
If \[\left( \cos x \right)^y = \left( \cos y \right)^x , \text{ find } \frac{dy}{dx}\] ?
If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
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 \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( 0, \frac{1}{\sqrt{2}} \right)\] ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
If f (x) = loge (loge x), then write the value of `f' (e)` ?
If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}\] ?
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
Find the second order derivatives of the following function x3 + tan x ?
If \[y = \frac{\log x}{x}\] show that \[\frac{d^2 y}{d x^2} = \frac{2 \log x - 3}{x^3}\] ?
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 \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write \[\frac{d^2 y}{d x^2}\] in terms of y ?
Let f(x) be a polynomial. Then, the second order derivative of f(ex) is
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
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to
Differentiate `log [x+2+sqrt(x^2+4x+1)]`
The number of road accidents in the city due to rash driving, over a period of 3 years, is given in the following table:
| Year | Jan-March | April-June | July-Sept. | Oct.-Dec. |
| 2010 | 70 | 60 | 45 | 72 |
| 2011 | 79 | 56 | 46 | 84 |
| 2012 | 90 | 64 | 45 | 82 |
Calculate four quarterly moving averages and illustrate them and original figures on one graph using the same axes for both.
