Advertisements
Advertisements
Question
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
Solution
\[\text{ We have, } \left( x + y \right)^2 = 2axy\]
Differentiating with respect to x, we get,
\[\Rightarrow \frac{d}{dx} \left( x + y \right)^2 = \frac{d}{dx}\left( 2axy \right)\]
\[ \Rightarrow 2\left( x + y \right)\frac{d}{dx}\left( x + y \right) = 2a\left[ x\frac{d y}{d x} + y\frac{d}{dx}\left( x \right) \right] \]
\[ \Rightarrow 2\left( x + y \right)\left[ 1 + \frac{d y}{d x} \right] = 2a\left[ x\frac{d y}{d x} + y\left( 1 \right) \right]\]
\[ \Rightarrow 2\left( x + y \right) + 2\left( x + y \right)\frac{d y}{d x} = 2ax\frac{d y}{d x} + 2ay\]
\[ \Rightarrow \frac{d y}{d x}\left[ 2\left( x + y \right) - 2ax \right] = 2ay - 2\left( x + y \right)\]
\[ \Rightarrow \frac{d y}{d x} = \frac{2\left[ ay - x - y \right]}{2\left[ x + y - ax \right]}\]
\[ \Rightarrow \frac{d y}{d x} = \left( \frac{ay - x - y}{x + y - ax} \right)\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles x2ex ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[\tan^{- 1} \left( e^x \right)\] ?
Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?
If \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
If \[y = e^x + e^{- x}\] prove that \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{\frac{1 - x}{2}} \right\}, 0 < x < 1\] ?
If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?
Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\] ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
If \[e^y = y^x ,\] prove that\[\frac{dy}{dx} = \frac{\left( \log y \right)^2}{\log y - 1}\] ?
Find the derivative of the function f (x) given by \[f\left( x \right) = \left( 1 + x \right) \left( 1 + x^2 \right) \left( 1 + x^4 \right) \left( 1 + x^8 \right)\] and hence find `f' (1)` ?
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 \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
Write the derivative of sinx with respect to cos x ?
Differentiate (log x)x with respect to log x ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\] \[x \in \left( - 1, 0 \right)\] ?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?
Differentiate \[\left( \cos x \right)^{\sin x }\] with respect to \[\left( \sin x \right)^{\cos x }\]?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
Find the second order derivatives of the following function tan−1 x ?
Find the second order derivatives of the following function log (log x) ?
If \[y = \frac{\log x}{x}\] show that \[\frac{d^2 y}{d x^2} = \frac{2 \log x - 3}{x^3}\] ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
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 ?
If y = etan x, then (cos2 x)y2 =
If x = f(t) cos t − f' (t) sin t and y = f(t) sin t + f'(t) cos t, then\[\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 =\]
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
