Advertisements
Advertisements
प्रश्न
Differentiate the following functions from first principles log cosec x ?
Advertisements
उत्तर
\[\text{Let} f\left( x \right) = \text{log cosecx}\]
\[ \Rightarrow f\left( x + h \right) = \text{log cosec}\left( x + h \right)\]
\[ \therefore \frac{d}{dx}\left\{ f\left( x \right) \right\} = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\text{log cosec}\left( x + h \right) - \log cosecx}{h}\]
\[ = \lim_{h \to 0} \frac{\log\left\{ \frac{cosec\left( x + h \right)}{cosecx} \right\}}{h}\]
\[ = \lim_{h \to 0} \frac{\log\left\{ 1 + \left( \frac{\sin x}{\sin\left( x + h \right)} - 1 \right) \right\}}{h}\]
\[ = \lim_{h \to 0} \left\{ \frac{\log\left\{ 1 + \left( \frac{\sin x - \sin\left( x + h \right)}{\sin\left( x + h \right)} \right) \right\}}{\left\{ \frac{\sin x - \sin\left( x + h \right)}{\sin\left( x + h \right)} \right\}} \right\}\frac{\left\{ \frac{\sin x - \sin\left( x + h \right)}{\sin\left( x + h \right)} \right\}}{h}\]
\[ = \lim_{h \to 0} \frac{2\cos\left( \frac{x + x + h}{2} \right)\sin\left( \frac{x - x - h}{2} \right)}{\sin\left( x + h \right)h} \left[ \because \lim_{x \to 0} \frac{\log\left( 1 + x \right)}{x} = 1 and \sin A - \sin B = 2\cos\left( \frac{A + B}{2} \right)\sin\left( \frac{A - B}{2} \right) \right]\]
\[ = \lim_{h \to 0} \frac{2\cos\left( \frac{2x + h}{2} \right)}{\sin\left( x + h \right) \left( - 2 \right)}\left\{ \frac{\sin\left( - \frac{h}{2} \right)}{- \frac{h}{2}} \right\} \left[ \because \lim_{x \to 0} \frac{\sin x}{x} = 1 \right]\]
\[ = - \cot x\]
\[ \therefore \frac{d}{dx}\left( \text{log cosec x} \right) = - \cot x\]
APPEARS IN
संबंधित प्रश्न
Differentiate sin (3x + 5) ?
Differentiate tan2 x ?
Differentiate (log sin x)2 ?
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\tan \left( e^{\sin x }\right)\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
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 \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sin x + \cos x}{\sqrt{2}} \right\}, - \frac{3 \pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{1 + \sqrt{1 - x^2}} \right\}, - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[xy = c^2\] ?
Find \[\frac{dy}{dx}\] in the following case \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?
If \[x \sqrt{1 + y} + y \sqrt{1 + x} = 0\] , prove that \[\left( 1 + x \right)^2 \frac{dy}{dx} + 1 = 0\] ?
If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
If \[e^x + e^y = e^{x + y} , \text{ prove that } \frac{dy}{dx} = - \frac{e^x \left( e^y - 1 \right)}{e^y \left( e^x - 1 \right)} or \frac{dy}{dx} + e^{y - x} = 0\] ?
Differentiate \[\left( 1 + \cos x \right)^x\] ?
Differentiate \[e^{\sin x }+ \left( \tan x \right)^x\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\sin x} + \left( \sin x \right)^x\] ?
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)` ?
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
If \[x = 2 \cos \theta - \cos 2 \theta \text{ and y} = 2 \sin \theta - \sin 2 \theta\], prove that \[\frac{dy}{dx} = \tan \left( \frac{3 \theta}{2} \right)\] ?
If \[x = a \left( \theta - \sin \theta \right) and, y = a \left( 1 + \cos \theta \right), \text { find } \frac{dy}{dx} \text{ at }\theta = \frac{\pi}{3} \] ?
Differentiate x2 with respect to x3
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\] ?
Differentiate\[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right)\] with respect to \[\sin^{-1} \left( \frac{2x}{1 + x^2} \right)\], If \[- 1 < x < 1, x \neq 0 .\] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
If `x = sin(1/2 log y)` show that (1 − x2)y2 − xy1 − a2y = 0.
If x = 2at, y = at2, where a is a constant, then find \[\frac{d^2 y}{d x^2} \text { at }x = \frac{1}{2}\] ?
If y = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
If xy = e(x – y), then show that `dy/dx = (y(x-1))/(x(y+1)) .`
f(x) = xx has a stationary point at ______.
