Advertisements
Advertisements
प्रश्न
If \[\left| x \right| < 1 \text{ and y} = 1 + x + x^2 + . . \] to ∞, then find the value of \[\frac{dy}{dx}\] ?
Advertisements
उत्तर
\[\text{ We have, y }= 1 + x + x^2 + . . . . to \ \infty \]
\[ \Rightarrow y = \frac{1}{1 - x} ...........[\because \text{ It is a G.P with first term }1\text{ and common ratio } x]\]
\[\Rightarrow \frac{dy}{dx} = \frac{d}{dx}\left( \frac{1}{1 - x} \right)\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{1}{\left( 1 - x \right)^2}\frac{d}{dx}\left( 1 - x \right)\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{1}{\left( 1 - x \right)^2}\left( - 1 \right)\]
\[ \Rightarrow \frac{dy}{dx} = \frac{1}{\left( 1 - x \right)^2}\]
APPEARS IN
संबंधित प्रश्न
Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
Differentiate \[\cos \left( \log x \right)^2\] ?
If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
If \[y = e^x + e^{- x}\] prove that \[\frac{dy}{dx} = \sqrt{y^2 - 4}\] ?
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
If \[y = \sin \left[ 2 \tan^{- 1} \left\{ \frac{\sqrt{1 - x}}{1 + x} \right\} \right], \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
If `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
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 \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?
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 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 \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?
If \[y = e^{x^{e^x}} + x^{e^{e^x}} + e^{x^{x^e}}\], prove that \[\frac{dy}{dx} = e^{x^{e^x}} \cdot x^{e^x} \left\{ \frac{e^x}{x} + e^x \cdot \log x \right\}+ x^{e^{e^x}} \cdot e^{e^x} \left\{ \frac{1}{x} + e^x \cdot \log x \right\} + e^{x^{x^e}} x^{x^e} \cdot x^{e - 1} \left\{ x + e \log x \right\}\]
Differentiate x2 with respect to x3
If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?
If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
If \[y = \sin^{- 1} x + \cos^{- 1} x\] ,find \[\frac{dy}{dx}\] ?
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
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 \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\] is equal to __________ .
Find the second order derivatives of the following function ex sin 5x ?
Find the second order derivatives of the following function log (log x) ?
If x = a (θ − sin θ), y = a (1 + cos θ) prove that, find \[\frac{d^2 y}{d x^2}\] ?
If x = cos θ, y = sin3 θ, prove that \[y\frac{d^2 y}{d x^2} + \left( \frac{dy}{dx} \right)^2 = 3 \sin^2 \theta\left( 5 \cos^2 \theta - 1 \right)\] ?
If x = sin t, y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
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 x = t2 and y = t3, find \[\frac{d^2 y}{d x^2}\] ?
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
If xy − loge y = 1 satisfies the equation \[x\left( y y_2 + y_1^2 \right) - y_2 + \lambda y y_1 = 0\]
If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
