Advertisements
Advertisements
Question
Find \[\frac{dy}{dx}\] in the following case \[x^5 + y^5 = 5 xy\] ?
Solution
\[\text{We have, }x^5 + y^5 = 5xy\]
Differentiating with respect to x, we get,
\[\frac{d}{dx}\left( x^5 \right) + \frac{d}{dx}\left( y^5 \right) = \frac{d}{dx}\left( 5xy \right)\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5\left[ x\frac{d y}{d x} + y\frac{d}{d x}\left( x \right) \right]\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5\left[ x\frac{d y}{d x} + y\left( 1 \right) \right]\]
\[ \Rightarrow 5 x^4 + 5 y^4 \frac{d y}{d x} = 5x\frac{d y}{d x} + 5y\]
\[ \Rightarrow 5 y^4 \frac{d y}{d x} - 5x\frac{d y}{d x} = 5y - 5 x^4 \]
\[ \Rightarrow 5\frac{d y}{d x}\left( y^4 - x \right) = 5\left( y - x^4 \right)\]
\[ \Rightarrow \frac{d y}{d x} = \frac{5\left( y - x^4 \right)}{5\left( y^4 - x \right)}\]
\[ \Rightarrow \frac{d y}{d x} = \frac{y - x^4}{y^4 - x}\]
APPEARS IN
RELATED QUESTIONS
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate the following functions from first principles e−x.
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?
Differentiate \[\cos^{- 1} \left\{ \sqrt{\frac{1 + x}{2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{2^{x + 1} \cdot 3^x}{1 + \left( 36 \right)^x} \right\}\] with respect to x ?
Find \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[\left( \log x \right)^x\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
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 \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
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 \[x^m y^n = 1\] , prove that \[\frac{dy}{dx} = - \frac{my}{nx}\] ?
If \[y = x \sin y\] , prove that \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?
Find \[\frac{dy}{dx}\] ,When \[x = a \left( 1 - \cos \theta \right) \text{ and } y = a \left( \theta + \sin \theta \right) \text{ at } \theta = \frac{\pi}{2}\] ?
If \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text { at } \left( 1/4, 1/4 \right)\text { is }\] _____________ .
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 \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .
Find the second order derivatives of the following function x3 log x ?
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
\[\text { If y } = x^n \left\{ a \cos\left( \log x \right) + b \sin\left( \log x \right) \right\}, \text { prove that } x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0 \] Disclaimer: There is a misprint in the question. It must be
\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] instead of 1
\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] ?
If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
If y = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
If x = t2, y = t3, then \[\frac{d^2 y}{d x^2} =\]
If \[y = \tan^{- 1} \left\{ \frac{\log_e \left( e/ x^2 \right)}{\log_e \left( e x^2 \right)} \right\} + \tan^{- 1} \left( \frac{3 + 2 \log_e x}{1 - 6 \log_e x} \right)\], then \[\frac{d^2 y}{d x^2} =\]
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
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 sin(log sin x) ?
