Advertisements
Advertisements
प्रश्न
Differentiate \[\tan^{- 1} \left( \frac{x - a}{x + a} \right)\] ?
Advertisements
उत्तर
\[\text{Let, y} = \tan^{- 1} \left( \frac{x - a}{x + a} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{\frac{x - a}{x}}{\frac{x + a}{x}} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{\frac{x}{x} - \frac{a}{x}}{\frac{x}{x} + \frac{a}{x}} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( \frac{1 - \frac{a}{x}}{1 + 1 \times \frac{a}{x}} \right)\]
\[ \Rightarrow y = \tan^{- 1} \left( 1 \right) - \tan^{- 1} \left( \frac{a}{x} \right)\]
Differentiate it with respect to x,
\[\frac{d y}{d x} = 0 - \frac{1}{1 + \left( \frac{a}{x} \right)^2}\frac{d}{dx}\left( \frac{a}{x} \right)\]
\[ \Rightarrow \frac{d y}{d x} = - \frac{x^2}{x^2 + a^2}\left( \frac{- a}{x^2} \right)\]
\[ \therefore \frac{d y}{d x} = \frac{a}{a^2 + x^2}\]
APPEARS IN
संबंधित प्रश्न
Differentiate log7 (2x − 3) ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[e^{ax} \sec x \tan 2x\] ?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + x}{1 - ax} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x^2 + y^2 \right)^2 = xy\] ?
If \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
If \[y^x + x^y + x^x = a^b\] ,find \[\frac{dy}{dx}\] ?
If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Find \[\frac{dy}{dx}\] , when \[x = \cos^{- 1} \frac{1}{\sqrt{1 + t^2}} \text{ and y } = \sin^{- 1} \frac{t}{\sqrt{1 + t^2}}, t \in R\] ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\] ?
If \[f'\left( 1 \right) = 2 \text { and y } = f \left( \log_e x \right), \text { find} \frac{dy}{dx} \text { at }x = e\] ?
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
If \[f\left( x \right) = \tan^{- 1} \sqrt{\frac{1 + \sin x}{1 - \sin x}}, 0 \leq x \leq \pi/2, \text{ then } f' \left( \pi/6 \right) \text{ is }\] _________ .
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 \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\] ____________ .
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 x3 + tan x ?
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 = (sin−1 x)2, prove that (1 − x2)
\[\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
If x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If y = a + bx2, a, b arbitrary constants, then
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
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 = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
f(x) = xx has a stationary point at ______.
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
