Advertisements
Advertisements
प्रश्न
Differentiate \[\sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\] ?
Advertisements
उत्तर
\[\text{ Let } y = \sqrt{\frac{a^2 - x^2}{a^2 + x^2}}\]
\[ \Rightarrow y = \left( \frac{a^2 - x^2}{a^2 + x^2} \right)^\frac{1}{2} \]
\[\text{Differentiate it with respect to x we get}, \]
\[\frac{d y}{d x} = \frac{d}{dx} \left( \frac{a^2 - x^2}{a^2 + x^2} \right)^\frac{1}{2} \]
\[ = \frac{1}{2} \left( \frac{a^2 - x^2}{a^2 + x^2} \right)^{\frac{1}{2} - 1} \times \frac{d}{dx}\left( \frac{a^2 - x^2}{a^2 + x^2} \right) \left[ \text{Using chain rule} \right]\]
\[ = \frac{1}{2} \left( \frac{a^2 - x^2}{a^2 + x^2} \right)^\frac{- 1}{2} \times \left\{ \frac{\left( a^2 + x^2 \right)\frac{d}{dx}\left( a^2 - x^2 \right) - \left( a^2 - x^2 \right)\frac{d}{dx}\left( a^2 + x^2 \right)}{\left( a^2 + x^2 \right)^2} \right\} \]
\[ = \frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 2x\left( a^2 + x^2 \right) - 2x\left( a^2 - x^2 \right)}{\left( a^2 + x^2 \right)^2} \right\}\]
\[ = \frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 2x a^2 - 2 x^3 - 2x a^2 + 2 x^3}{\left( a^2 + x^2 \right)^2} \right\}\]
\[ = \frac{1}{2} \left( \frac{a^2 + x^2}{a^2 - x^2} \right)^\frac{1}{2} \left\{ \frac{- 4x a^2}{\left( a^2 + x^2 \right)^2} \right\}\]
\[ = \frac{- 2x a^2}{\sqrt{a^2 - x^2} \left( a^2 + x^2 \right)^\frac{3}{2}}\]
\[So, \frac{d}{dx}\left( \sqrt{\frac{a^2 - x^2}{a^2 + x^2}} \right) = \frac{- 2 a^2 x}{\sqrt{a^2 - x^2} \left( a^2 + x^2 \right)^\frac{3}{2}}\]
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−x.
Differentiate the following functions from first principles eax+b.
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\sin^2 \left\{ \log \left( 2x + 3 \right) \right\}\] ?
Differentiate \[3 e^{- 3x} \log \left( 1 + x \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{2^{x + 1}}{1 - 4^x} \right), - \infty < x < 0\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[\left( \sin x \right)^{\log x}\] ?
Differentiate \[x^{\sin^{- 1} x}\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\]
\[y = x^x + x^{1/x}\] ?
If \[y^x = e^{y - x}\] ,prove that \[\frac{dy}{dx} = \frac{\left( 1 + \log y \right)^2}{\log y}\] ?
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
If \[x = \frac{\sin^3 t}{\sqrt{\cos 2 t}}, y = \frac{\cos^3 t}{\sqrt{\cos t 2 t}}\] , find\[\frac{dy}{dx}\] ?
Write the derivative of sinx with respect to cos 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 \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[u = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) \text{ and v} = \tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] where \[- 1 < x < 1\], then write the value of \[\frac{du}{dv}\] ?
If f (x) = logx2 (log x), the `f' (x)` at x = e is ____________ .
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
For the curve \[\sqrt{x} + \sqrt{y} = 1, \frac{dy}{dx}\text { at } \left( 1/4, 1/4 \right)\text { is }\] _____________ .
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
If \[f\left( x \right) = \left| x - 3 \right| \text { and }g\left( x \right) = fof \left( x \right)\] is equal to __________ .
If y = sin (sin x), prove that \[\frac{d^2 y}{d x^2} + \tan x \cdot \frac{dy}{dx} + y \cos^2 x = 0\] ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
Find \[\frac{d^2 y}{d x^2}\] where \[y = \log \left( \frac{x^2}{e^2} \right)\] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If y = a cos (loge x) + b sin (loge x), then x2 y2 + xy1 =
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
