Advertisements
Advertisements
प्रश्न
Differentiate the following functions from first principles log cos x ?
उत्तर
\[\text{ Let } f\left( x \right) = \log \cos x\]
\[ \Rightarrow f\left( x + h \right) = \log \cos\left( x + h \right)\]
\[ \therefore \frac{d}{dx}\left\{ f\left( x \right) \right\} = \lim_{h \to 0} \frac{f\left( x + h \right) = f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\log \cos\left( x + h \right) - \log \cos x}{h}\]
\[ = \lim_{h \to 0} \frac{\log\frac{\cos\left( x + h \right)}{\cos x}}{h} \left[ \because \log A - \log B = \log\left( \frac{A}{B} \right) \right]\]
\[ = \lim_{h \to 0} \frac{\log\left[ 1 + \left\{ \frac{\cos\left( x + h \right)}{\cos x} - 1 \right\} \right]}{h}\]
\[ = \lim_{h \to 0} \frac{\log\left\{ 1 + \frac{\cos\left( x + h \right) - \cos x}{\cos x} \right\}}{\left\{ \frac{\cos\left( x + h \right) - \cos x}{\cos x} \right\}} \times \lim_{h \to 0} \left\{ \frac{\cos\left( x + h \right) - \cos x}{\cos x} \right\}\]
\[ = 1 \times \lim_{h \to 0} \frac{\cos\left( x + h \right) - \cos x}{\cos x \times h} \left[ \because \lim_{x \to 0} \frac{\log\left( 1 + x \right)}{x} = 1 \right]\]
\[ = \lim_{h \to 0} \frac{- 2\sin\left( \frac{x + h + x}{2} \right)\sin\left( \frac{x + h - x}{2} \right)}{\cos x \times h}\]
\[ = - 2 \lim_{h \to 0} \frac{\sin\left( \frac{2x + h}{2} \right) \times \sin\left( \frac{h}{2} \right)}{2\cos x \times \left( \frac{h}{2} \right)}\]
\[ = \frac{- 2\sin x}{2\cos x} \left[ \because \lim_{x \to 0} \frac{\sin x}{x} = 1 \right]\]
\[ = - \tan x\]
\[So, \frac{d}{dx}\left( \log \cos x \right) = - \tan x\]
APPEARS IN
संबंधित प्रश्न
Differentiate log7 (2x − 3) ?
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[e^\sqrt{\cot x}\] ?
Differentiate \[\frac{e^x \log x}{x^2}\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[e^{\tan^{- 1}} \sqrt{x}\] ?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
If \[y = \frac{1}{2} \log \left( \frac{1 - \cos 2x }{1 + \cos 2x} \right)\] , prove that \[\frac{ dy }{ dx } = 2 \text{cosec }2x \] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?
If \[y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty\] prove that \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
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 \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
If \[y = \tan^{- 1} \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \right), \text{find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
If \[xy = 1\] prove that \[\frac{dy}{dx} + y^2 = 0\] ?
If \[xy \log \left( x + y \right) = 1\] ,Prove that \[\frac{dy}{dx} = - \frac{y \left( x^2 y + x + y \right)}{x \left( x y^2 + x + y \right)}\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[\left( \sin x \right)^{\log x}\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
If \[x^y \cdot y^x = 1\] , prove that \[\frac{dy}{dx} = - \frac{y \left( y + x \log y \right)}{x \left( y \log x + x \right)}\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
Differentiate \[\sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { if } 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{x - 1}{x + 1} \right)\] with respect to \[\sin^{- 1} \left( 3x - 4 x^3 \right), \text { if }- \frac{1}{2} < x < \frac{1}{2}\] ?
If \[x = a \left( \theta + \sin \theta \right), y = a \left( 1 + \cos \theta \right), \text{ find} \frac{dy}{dx}\] ?
If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
If \[y = \tan^{- 1} \left( \frac{\sin x + \cos x}{\cos x - \sin x} \right), \text { then } \frac{dy}{dx}\] is equal to ___________ .
Find the second order derivatives of the following function x3 + tan x ?
Find the second order derivatives of the following function sin (log x) ?
If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?
If x = a sec θ, y = b tan θ, prove that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If \[y = e^{2x} \left( ax + b \right)\] show that \[y_2 - 4 y_1 + 4y = 0\] ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
If y log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
If x = at2, y = 2 at, then \[\frac{d^2 y}{d x^2} =\]
If y = sin (m sin−1 x), then (1 − x2) y2 − xy1 is equal to
If logy = tan–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`
