Advertisements
Advertisements
Question
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Solution
\[\text{ Let } y = \sqrt{\frac{1 - x^2}{1 + x^2}}\]
\[ \Rightarrow y = \left( \frac{1 - x^2}{1 + x^2} \right)^\frac{1}{2} \]
\[\text{ Differentiate it with respect to x we get }, \]
\[\frac{d y}{d x} = \frac{d}{dx} \left( \frac{1 - x^2}{1 + x^2} \right)^\frac{1}{2} \]
\[ = \frac{1}{2} \left( \frac{1 - x^2}{1 + x^2} \right)^{\frac{1}{2} - 1} \times \frac{d}{dx}\left( \frac{1 - x^2}{1 + x^2} \right) \left[ \text{ Using chain rule } \right]\]
\[ = \frac{1}{2} \left( \frac{1 - x^2}{1 + x^2} \right)^\frac{- 1}{2} \times \left\{ \frac{\left( 1 + x^2 \right)\frac{d}{dx}\left( 1 - x^2 \right) - \left( 1 - x^2 \right)\frac{d}{dx}\left( 1 + x^2 \right)}{\left( 1 + x^2 \right)^2} \right\} \]
\[ = \frac{1}{2} \left( \frac{1 + x^2}{1 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 2x\left( 1 + x^2 \right) - 2x\left( 1 - x^2 \right)}{\left( 1 + x^2 \right)^2} \right\}\]
\[ = \frac{1}{2} \left( \frac{1 + x^2}{1 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 2x - 2 x^3 - 2x + 2 x^3}{\left( 1 + x^2 \right)^2} \right\}\]
\[ = \frac{1}{2} \left( \frac{1 + x^2}{1 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 4x}{\left( 1 + x^2 \right)^2} \right\}\]
\[ = \frac{- 2x}{\sqrt{1 - x^2} \left( 1 + x^2 \right)^\frac{3}{2}}\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles e−x.
Differentiate \[3^{x^2 + 2x}\] ?
Differentiate \[\log \left( \frac{\sin x}{1 + \cos x} \right)\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\log \left( \frac{x^2 + x + 1}{x^2 - x + 1} \right)\] ?
Differentiate \[x \sin 2x + 5^x + k^k + \left( \tan^2 x \right)^3\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
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 \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
If \[y \sqrt{1 - x^2} + x \sqrt{1 - y^2} = 1\] ,prove that \[\frac{dy}{dx} = - \sqrt{\frac{1 - y^2}{1 - x^2}}\] ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
Differentiate \[{10}^\left( {10}^x \right)\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?
If \[y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sec^2 x}{2 y - 1}\] ?
Find \[\frac{dy}{dx}\] when \[x = \frac{2 t}{1 + t^2} \text{ and } y = \frac{1 - t^2}{1 + t^2}\] ?
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 - x}{1 + x} \right)\] with respect to \[\sqrt{1 - x^2},\text {if} - 1 < x < 1\] ?
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If \[\sqrt{1 - x^6} + \sqrt{1 - y^6} = a^3 \left( x^3 - y^3 \right)\] then \[\frac{dy}{dx}\] is equal to ____________ .
Find the second order derivatives of the following function x3 log x ?
Find the second order derivatives of the following function tan−1 x ?
If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
If y = etan x, then (cos2 x)y2 =
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
If y = xn−1 log x then x2 y2 + (3 − 2n) xy1 is equal to
If logy = tan–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`
