Advertisements
Advertisements
प्रश्न
If \[x^{13} y^7 = \left( x + y \right)^{20}\] prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
Advertisements
उत्तर
\[\text{ We have,} x^{13} y^7 = \left( x + y \right)^{20} \]
Taking log on both sides,
\[\log\left( x^{13} y^7 \right) = \log \left( x + y \right)^{20} \]
\[ \Rightarrow 13\log x + 7\log y = 20\log\left( x + y \right)\]
Differentiating with respect to x using chain rule,
\[13\frac{d}{dx}\left( \log x \right) + 7\frac{d}{dx}\left( \log y \right) = 20\frac{d}{dx}\log\left( x + y \right)\]
\[ \Rightarrow \frac{13}{x} + \frac{7}{y}\frac{dy}{dx} = \frac{20}{x + y}\frac{d}{dx}\left( x + y \right)\]
\[ \Rightarrow \frac{13}{x} + \frac{7}{y}\frac{dy}{dx} = \frac{20}{x + y}\left[ 1 + \frac{dy}{dx} \right]\]
\[ \Rightarrow \frac{7}{y}\frac{dy}{dx} - \frac{20}{x + y}\frac{dy}{dx} = \frac{20}{x + y} - \frac{13}{x}\]
\[ \Rightarrow \frac{dy}{dx}\left[ \frac{7}{y} - \frac{20}{x + y} \right] = \frac{20}{x + y} - \frac{13}{x}\]
\[ \Rightarrow \frac{dy}{dx}\left[ \frac{7\left( x + y \right) - 20y}{y\left( x + y \right)} \right] = \left[ \frac{20x - 13\left( x + y \right)}{x\left( x + y \right)} \right]\]
\[ \Rightarrow \frac{dy}{dx} = \left[ \frac{20x - 13x - 13y}{x\left( x + y \right)} \right]\left[ \frac{y\left( x + y \right)}{7x + 7y - 20y} \right]\]
\[ \Rightarrow \frac{dy}{dx} = \frac{y}{x}\left( \frac{7x - 13y}{7x - 13y} \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{y}{x}\]
APPEARS IN
संबंधित प्रश्न
Differentiate sin2 (2x + 1) ?
Differentiate `2^(x^3)` ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[\sqrt{\tan^{- 1} \left( \frac{x}{2} \right)}\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\sin^{- 1} \left\{ \sqrt{1 - x^2} \right\}, 0 < x < 1\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{4x}{1 - 4 x^2} \right), - \frac{1}{2} < x < \frac{1}{2}\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
Differentiate \[x^{\cos^{- 1} x}\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Find \[\frac{dy}{dx}\] \[y = e^{3x} \sin 4x \cdot 2^x\] ?
Find \[\frac{dy}{dx}\], When \[x = a \left( \theta + \sin \theta \right) \text{ and } y = a \left( 1 - \cos \theta \right)\] ?
Find \[\frac{dy}{dx}\], when \[x = a \left( \cos \theta + \theta \sin \theta \right) \text{ and }y = a \left( \sin \theta - \theta \cos \theta \right)\] ?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
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 (log x)x with respect to log x ?
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 \[\tan^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right)\] with respect to \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right), \text { if } - \frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\] ?
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 \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .
Find the second order derivatives of the following function log (sin x) ?
If y = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
If \[y = e^{2x} \left( ax + b \right)\] show that \[y_2 - 4 y_1 + 4y = 0\] ?
If y = ex (sin x + cos x) prove that \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\] ?
If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?
If y = x + ex, find \[\frac{d^2 x}{d y^2}\] ?
If \[y = \left| \log_e x \right|\] 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 x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If f(x) = (cos x + i sin x) (cos 2x + i sin 2x) (cos 3x + i sin 3x) ...... (cos nx + i sin nx) and f(1) = 1, then f'' (1) is equal to
Find the minimum value of (ax + by), where xy = c2.
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
