Advertisements
Advertisements
प्रश्न
If \[\sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \text { log a then } \frac{dy}{dx}\] is equal to _____________ .
विकल्प
\[\frac{x^2 - y^2}{x^2 + y^2}\]
`y/x`
`x/y`
none of these
उत्तर
\[\frac{y}{x}\]
\[\text { We have }, \sin^{- 1} \left( \frac{x^2 - y^2}{x^2 + y^2} \right) = \log a\]
\[ \Rightarrow \frac{x^2 - y^2}{x^2 + y^2} = \sin \log a\]
\[\Rightarrow \frac{\left( x^2 + y^2 \right)\left( 2x - 2y\frac{dy}{dx} \right) - \left( x^2 - y^2 \right)\left( 2x + 2y\frac{dy}{dx} \right)}{\left( x^2 + y^2 \right)^2} = 0\]
\[ \Rightarrow \frac{2 x^3 - 2 x^2 y\frac{dy}{dx} + 2x y^2 - 2 y^3 \frac{dy}{dx} - 2 x^3 - 2 x^2 y\frac{dy}{dx} + 2x y^2 + 2 y^3 \frac{dy}{dx}}{\left( x^2 + y^2 \right)^2} = 0\]
\[ \Rightarrow - 4 x^2 y\frac{dy}{dx} + 4x y^2 = 0\]
\[ \Rightarrow - 4 x^2 y\frac{dy}{dx} = - 4x y^2 \]
\[ \Rightarrow \frac{dy}{dx} = \frac{4x y^2}{4 x^2 y}\]
\[ \therefore \frac{dy}{dx} = \frac{y}{x}\]
APPEARS IN
संबंधित प्रश्न
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `cos^(-1)(1/sqrt3)`
If the function f(x)=2x3−9mx2+12m2x+1, where m>0 attains its maximum and minimum at p and q respectively such that p2=q, then find the value of m.
Differentiate etan x ?
Differentiate \[\sqrt{\frac{1 + x}{1 - x}}\] ?
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[\log \left( 3x + 2 \right) - x^2 \log \left( 2x - 1 \right)\] ?
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
Differentiate \[\sin^{- 1} \left( \frac{x}{\sqrt{x^2 + a^2}} \right)\] ?
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\sin^{- 1} \left( 2 x^2 - 1 \right), 0 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x \in R\] ?
If \[y = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), x > 0\] ,prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x^2 + y^2 \right)^2 = xy\] ?
If \[y \sqrt{x^2 + 1} = \log \left( \sqrt{x^2 + 1} - x \right)\] ,Show that \[\left( x^2 + 1 \right) \frac{dy}{dx} + xy + 1 = 0\] ?
Differentiate \[x^{\sin x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = x^x + \left( \sin x \right)^x\] ?
If \[\left( \sin x \right)^y = \left( \cos y \right)^x ,\], prove that \[\frac{dy}{dx} = \frac{\log \cos y - y cot x}{\log \sin x + x \tan y}\] ?
If \[\left( \cos x \right)^y = \left( \tan y \right)^x\] , prove that \[\frac{dy}{dx} = \frac{\log \tan y + y \tan x}{ \log \cos x - x \sec y \ cosec\ y }\] ?
If \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?
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 \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?
If \[x = e^{\cos 2 t} \text{ and y }= e^{\sin 2 t} ,\] prove that \[\frac{dy}{dx} = - \frac{y \log x}{x \log y}\] ?
If \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?
If \[x = 10 \left( t - \sin t \right), y = 12 \left( 1 - \cos t \right), \text { find } \frac{dy}{dx} .\] ?
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\tan^{- 1} \left( \frac{2 x}{1 - x^2} \right), \text{ if } - 1 < x < 1\] ?
If \[\pi \leq x \leq 2\pi \text { and y } = \cos^{- 1} \left( \cos x \right), \text { find } \frac{dy}{dx}\] ?
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
If \[y = \frac{1}{1 + x^{a - b} +^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}}\] then \[\frac{dy}{dx}\] is equal to ______________ .
Find the second order derivatives of the following function tan−1 x ?
If x = a (θ + sin θ), y = a (1 + cos θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{a}{y^2}\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
If y = |x − x2|, then find \[\frac{d^2 y}{d x^2}\] ?
If y = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
If y2 = ax2 + bx + c, then \[y^3 \frac{d^2 y}{d x^2}\] is
If logy = tan–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`
f(x) = 3x2 + 6x + 8, x ∈ R
