Advertisements
Advertisements
प्रश्न
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
Advertisements
उत्तर
\[\text{ We have, y } = \frac{e^{ax} \sec x \log x}{\sqrt{1 - 2x}} . . . \left( i \right)\]
\[ \Rightarrow y = \frac{e^{ax} \sec x \log x}{\left( 1 - 2x \right)^\frac{1}{2}}\]
Taking log on both sides
\[\log y = \log e^{ax} + logsec x + \log \log x - \frac{1}{2}\log\left( 1 - 2x \right) \]
\[ \Rightarrow \log y = ax + \log\left( \sec x \right) + \log\left( \log x \right) - \frac{1}{2}\log\left( 1 - 2x \right) \]
Differentiating with respect to x using chain rule,
\[\frac{1}{y}\frac{dy}{dx} = \frac{d}{dx}\left( ax \right) + \frac{d}{dx}\left( \log \sec x \right) + \frac{d}{dx}\left( \log \log x \right) - \frac{1}{2}\log\left( 1 - 2x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = a + \frac{1}{\sec x}\frac{d}{dx}\left( \sec x \right) + \frac{1}{\log x}\frac{d}{dx}\left( \log x \right) - \frac{1}{2}\left( \frac{1}{1 - 2x} \right)\frac{d}{dx}\left( 1 - 2x \right)\]
\[ \Rightarrow \frac{1}{y}\frac{dy}{dx} = a + \frac{\sec x \tan x}{\sec x} + \frac{1}{\left( \log x \right)}\left( \frac{1}{x} \right) - \frac{1}{2}\left( \frac{1}{1 - 2x} \right)\left( - 2 \right)\]
\[ \Rightarrow \frac{dy}{dx} = y\left[ a + \tan x + \frac{1}{x \log x} + \frac{1}{1 - 2x} \right]\]
\[ \Rightarrow \frac{dy}{dx} = \frac{e^{ax} \sec x \log x}{\sqrt{1 - 2x}}\left[ a + \tan x + \frac{1}{x \log x} + \frac{1}{1 - 2x} \right] \left[ \text{ Using equation }\left( i \right) \right]\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles log cos x ?
Differentiate sin2 (2x + 1) ?
Differentiate log7 (2x − 3) ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[\log \left( x + \sqrt{x^2 + 1} \right)\] ?
Differentiate \[\cos^{- 1} \left\{ 2x\sqrt{1 - x^2} \right\}, \frac{1}{\sqrt{2}} < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + a^2 x^2} - 1}{ax} \right), x \neq 0\] ?
Differentiate \[\cos^{- 1} \left( \frac{1 - x^{2n}}{1 + x^{2n}} \right), < x < \infty\] ?
Find \[\frac{dy}{dx}\] in the following case \[\left( x + y \right)^2 = 2axy\] ?
Find \[\frac{dy}{dx}\] in the following case \[e^{x - y} = \log \left( \frac{x}{y} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[\sin xy + \cos \left( x + y \right) = 1\] ?
If \[x y^2 = 1,\] prove that \[2\frac{dy}{dx} + y^3 = 0\] ?
If \[y = \left\{ \log_{\cos x} \sin x \right\} \left\{ \log_{\sin x} \cos x \right\}^{- 1} + \sin^{- 1} \left( \frac{2x}{1 + x^2} \right), \text{ find } \frac{dy}{dx} \text{ at }x = \frac{\pi}{4}\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
If \[y = \sin \left( x^x \right)\] prove that \[\frac{dy}{dx} = \cos \left( x^x \right) \cdot x^x \left( 1 + \log x \right)\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
If \[y = \left( \sin x - \cos x \right)^{\sin x - \cos x} , \frac{\pi}{4} < x < \frac{3\pi}{4}, \text{ find} \frac{dy}{dx}\] ?
\[y = \left( \sin x \right)^{\left( \sin x \right)^{\left( \sin x \right)^{. . . \infty}}} \],prove that \[\frac{y^2 \cot x}{\left( 1 - y \log \sin x \right)}\] ?
If \[y = \left( \tan x \right)^{\left( \tan x \right)^{\left( \tan x \right)^{. . . \infty}}}\], prove that \[\frac{dy}{dx} = 2\ at\ x = \frac{\pi}{4}\] ?
If \[x = \cos t \text{ and y } = \sin t,\] prove that \[\frac{dy}{dx} = \frac{1}{\sqrt{3}} \text { at } t = \frac{2 \pi}{3}\] ?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
\[\text { If }x = \cos t\left( 3 - 2 \cos^2 t \right), y = \sin t\left( 3 - 2 \sin^2 t \right) \text { find the value of } \frac{dy}{dx}\text{ at }t = \frac{\pi}{4}\] ?
Differentiate \[\cos^{- 1} \left( 4 x^3 - 3x \right)\] with respect to \[\tan^{- 1} \left( \frac{\sqrt{1 - x^2}}{x} \right), \text{ if }\frac{1}{2} < x < 1\] ?
If \[f'\left( x \right) = \sqrt{2 x^2 - 1} \text { and y } = f \left( x^2 \right)\] then find \[\frac{dy}{dx} \text { at } x = 1\] ?
If \[y = x \left| x \right|\] , find \[\frac{dy}{dx} \text{ for } x < 0\] ?
If \[y = \log_a x, \text{ find } \frac{dy}{dx} \] ?
If \[f\left( x \right) = \log \left\{ \frac{u \left( x \right)}{v \left( x \right)} \right\}, u \left( 1 \right) = v \left( 1 \right) \text{ and }u' \left( 1 \right) = v' \left( 1 \right) = 2\] , then find the value of `f' (1)` ?
Differential coefficient of sec(tan−1 x) is ______.
If \[y = \left( 1 + \frac{1}{x} \right)^x , \text{then} \frac{dy}{dx} =\] ____________.
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .
The derivative of \[\cos^{- 1} \left( 2 x^2 - 1 \right)\] with respect to \[\cos^{- 1} x\] is ___________ .
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 ex sin 5x ?
Find the second order derivatives of the following function x cos x ?
\[\text{ If x } = a\left( \cos t + \log \tan\frac{t}{2} \right) \text { and y } = a\left( \sin t \right), \text { evaluate } \frac{d^2 y}{d x^2} \text { at t } = \frac{\pi}{3} \] ?
If y = x + ex, find \[\frac{d^2 x}{d y^2}\] ?
If x = t2, y = t3, then \[\frac{d^2 y}{d x^2} =\]
