Advertisements
Advertisements
प्रश्न
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
विकल्प
(xy1 − y)2
(1 + y)2
\[\left( \frac{y - x y_1}{y_1} \right)^2\]
none of these
उत्तर
(a) (xy1 − y)2
Here,
\[y = \log_e \left( \frac{x}{a + bx} \right)^x \]
\[ \Rightarrow y = x \log_e \left( \frac{x}{a + bx} \right) \]
\[ \Rightarrow y_1 = \log_e \left( \frac{x}{a + bx} \right) + x \times \frac{a + bx}{x}\left( \frac{1}{a + bx} - \frac{bx}{\left( a + bx \right)^2} \right)\]
\[ \Rightarrow y_1 = \log_e \left( \frac{x}{a + bx} \right) + \left( \frac{a}{a + bx} \right) . . . \left( 1 \right)\]
\[ \Rightarrow y_1 = \frac{y}{x} + \left( \frac{a}{a + bx} \right) \left[ \because y = x \log_e \left( \frac{x}{a + bx} \right) \right]\]
\[ \Rightarrow \frac{x y_1 - y}{x} = \frac{a}{a + bx} . . . \left( 2 \right)\]
\[\text{Differentiating } \left( 1 \right) \text { we get }, \]
\[ y_2 = \frac{a + bx}{x}\left( \frac{a + bx - bx}{\left( a + bx \right)^2} \right) - \frac{ba}{\left( a + bx \right)^2}\]
\[ \Rightarrow y_2 = \frac{a}{x\left( a + bx \right)} - \frac{ba}{\left( a + bx \right)^2}\]
\[ \Rightarrow y_2 = \frac{a\left( a + bx \right) - abx}{x \left( a + bx \right)^2}\]
\[ \Rightarrow y_2 = \frac{a^2}{x \left( a + bx \right)^2}\]
\[ \Rightarrow y_2 = \frac{\left( x y_1 - y \right)^2}{x^3} \left[ \text { Using }
\left( 2 \right) \right]\]
\[ \Rightarrow x^3 y_2 = \left( x y_1 - y \right)^2 \]
APPEARS IN
संबंधित प्रश्न
If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\], prove that \[2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
If \[x \sin \left( a + y \right) + \sin a \cos \left( a + y \right) = 0\] Prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
If `y=(sinx)^x + sin^-1 sqrtx "then find" dy/dx`
If \[y = x \sin \left( a + y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sin^2 \left( a + y \right)}{\sin \left( a + y \right) - y \cos \left( a + y \right)}\] ?
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 \[x = \frac{1 + \log t}{t^2}, y = \frac{3 + 2\log t}{t}, \text { find } \frac{dy}{dx}\] ?
Differentiate (log x)x with respect to log x ?
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 \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
The differential coefficient of f (log x) w.r.t. x, where f (x) = log x is ___________ .
For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text { at } \left( 1/4, 1/4 \right)\text { is }\] _____________ .
Let \[\cup = \sin^{- 1} \left( \frac{2 x}{1 + x^2} \right) \text { and }V = \tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text { then } \frac{d \cup}{dV} =\] ____________ .
If \[\sin y = x \cos \left( a + y \right), \text { then } \frac{dy}{dx}\] is equal to ______________ .
Find the second order derivatives of the following function x3 + tan x ?
Find the second order derivatives of the following function tan−1 x ?
If y = x + tan x, show that \[\cos^2 x\frac{d^2 y}{d x^2} - 2y + 2x = 0\] ?
If x = a (1 − cos3 θ), y = a sin3 θ, prove that \[\frac{d^2 y}{d x^2} = \frac{32}{27a} \text { at } \theta = \frac{\pi}{6}\] ?
If y log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
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 = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
