Advertisements
Advertisements
Question
If \[x^{16} y^9 = \left( x^2 + y \right)^{17}\] ,prove that \[x\frac{dy}{dx} = 2 y\] ?
Solution
\[\text{ We have}, x^{16} y^9 = \left( x^2 + y \right)^{17} \]
Taking log on both sides,
\[\log\left( x^{16} y^9 \right) = \log \left( x^2 + y \right)^{17} \]
\[ \Rightarrow 16\log x + 9\log y = 17\log\left( x^2 + y \right)\]
Differentiating with respect to x using chain rule,
\[16\frac{d}{dx}\left( \log x \right) + 9\frac{d}{dx}\left( \log y \right) = 17\frac{d}{dx}\log\left( x^2 + y \right)\]
\[ \Rightarrow \frac{16}{x} + \frac{9}{y}\frac{dy}{dx} = \frac{17}{x^2 + y}\frac{d}{dx}\left( x^2 + y \right)\]
\[ \Rightarrow \frac{16}{x} + \frac{9}{y}\frac{dy}{dx} = \frac{17}{x^2 + y}\left[ 2x + \frac{dy}{dx} \right]\]
\[ \Rightarrow \frac{9}{y}\frac{dy}{dx} - \frac{17}{x^2 + y}\frac{dy}{dx} = \frac{34x}{x^2 + y} - \frac{16}{x}\]
\[ \Rightarrow \frac{dy}{dx}\left[ \frac{9}{y} - \frac{17}{x^2 + y} \right] = \frac{34x}{x^2 + y} - \frac{16}{x}\]
\[ \Rightarrow \frac{dy}{dx}\left[ \frac{9\left( x^2 + y \right) - 17y}{y\left( x^2 + y \right)} \right] = \left[ \frac{34 x^2 - 16\left( x^2 + y \right)}{x\left( x^2 + y \right)} \right]\]
\[ \Rightarrow \frac{dy}{dx}\left[ \frac{9 x^2 + 9y - 17y}{y\left( x^2 + y \right)} \right] = \left[ \frac{34 x^2 - 16 x^2 - 16y}{x\left( x^2 + y \right)} \right]\]
\[ \Rightarrow \frac{dy}{dx}\left[ \frac{9 x^2 - 8y}{y\left( x^2 + y \right)} \right] = \left[ \frac{18 x^2 - 16y}{x\left( x^2 + y \right)} \right]\]
\[ \Rightarrow \frac{dy}{dx} = \frac{y}{x}\left[ \frac{2\left( 9 x^2 - 8y \right)}{9 x^2 - 8y} \right]\]
\[ \Rightarrow \frac{dy}{dx} = \frac{2y}{x}\]
\[ \Rightarrow x\frac{dy}{dx} = 2y\]
APPEARS IN
RELATED QUESTIONS
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate the following functions from first principles ecos x.
Differentiate `2^(x^3)` ?
Differentiate \[3^{e^x}\] ?
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
If \[y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]prove that \[\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}\] ?
Prove that \[\frac{d}{dx} \left\{ \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{- 1} \frac{x}{a} \right\} = \sqrt{a^2 - x^2}\] ?
Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?
If \[y = \cot^{- 1} \left\{ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right\}\], show that \[\frac{dy}{dx}\] is independent of x. ?
If \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?
Differentiate \[x^{1/x}\] with respect to x.
Differentiate \[\left( \sin^{- 1} x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
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 \[e^x + e^y = e^{x + y}\] , prove that
\[\frac{dy}{dx} + e^{y - x} = 0\] ?
If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .
If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 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 y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
