Advertisements
Advertisements
प्रश्न
Differentiate \[\frac{x^2 + 2}{\sqrt{\cos x}}\] ?
Advertisements
उत्तर
\[\text{Let }y = \frac{x^2 + 2}{\sqrt{\cos x}}\]
Differentiate it with respect to x we get,
\[\frac{d y}{d x} = \frac{\sqrt{\cos x}\frac{d}{dx}\left( x^2 + 2 \right) - \left( x^2 + 2 \right)\frac{d}{dx}\left( \sqrt{\cos x} \right)}{\left( \sqrt{\cos x} \right)^2} \left[ \text{Using quotient rule and chain rule} \right]\]
\[ = \frac{2x\sqrt{\cos x} - \left( x^2 + 2 \right)\left( - \frac{1}{2}\frac{\sin x}{\sqrt{\cos x}} \right)}{\cos x}\]
\[ = \frac{2x\sqrt{\cos x} + \frac{\left( x^2 + 2 \right)\sin x}{2\sqrt{\cos x}}}{\cos x}\]
\[ = \frac{4x \cos x + \left( x^2 + 2 \right)\sin x}{2 \left( \cos x \right)^\frac{3}{2}}\]
\[ = \frac{2x}{\sqrt{\cos x}} + \frac{1}{2}\frac{\left( x^2 + 2 \right)\sin x}{\left( \cos x \right)^\frac{3}{2}}\]
\[ = \frac{1}{\sqrt{\cos x}}\left\{ 2x + \frac{1}{2}\frac{\left( x^2 + 2 \right)\sin x}{\cos x} \right\}\]
\[ = \frac{1}{\sqrt{\cos x}}\left\{ 2x + \frac{\left( x^2 + 2 \right)\tan x}{2} \right\}\]
\[So, \frac{d}{dx}\left( \frac{x^2 + 2}{\sqrt{\cos x}} \right) = \frac{1}{\sqrt{\cos x}}\left\{ 2x + \frac{\left( x^2 + 2 \right)\tan x}{2} \right\}\]
APPEARS IN
संबंधित प्रश्न
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate the following functions from first principles ecos x.
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[\tan^{- 1} \left( e^x \right)\] ?
If \[y = \log \left\{ \sqrt{x - 1} - \sqrt{x + 1} \right\}\] ,show that \[\frac{dy}{dx} = \frac{- 1}{2\sqrt{x^2 - 1}}\] ?
If \[y = \log \sqrt{\frac{1 + \tan x}{1 - \tan x}}\] prove that \[\frac{dy}{dx} = \sec 2x\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
If \[y = \sqrt{a^2 - x^2}\] prove that \[y\frac{dy}{dx} + x = 0\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right) + \sec^{- 1} \left( \frac{1 + x^2}{1 - x^2} \right), 0 < x < 1,\] prove that \[\frac{dy}{dx} = \frac{4}{1 + x^2}\] ?
Find \[\frac{dy}{dx}\] in the following case \[x^{2/3} + y^{2/3} = a^{2/3}\] ?
If \[\sec \left( \frac{x + y}{x - y} \right) = a\] Prove that \[\frac{dy}{dx} = \frac{y}{x}\] ?
If \[\sin^2 y + \cos xy = k,\] find \[\frac{dy}{dx}\] at \[x = 1 , \] \[y = \frac{\pi}{4} .\]
Differentiate \[x^{\cos^{- 1} x}\] ?
Differentiate \[\left( \sin x \right)^{\cos x}\] ?
Differentiate \[\left( \tan x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = \frac{e^{ax} \cdot \sec x \cdot \log x}{\sqrt{1 - 2x}}\] ?
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 \[xy = e^{x - y} , \text{ find } \frac{dy}{dx}\] ?
Find \[\frac{dy}{dx}\],when \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
Given \[f\left( x \right) = 4 x^8 , \text { then }\] _________________ .
If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .
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 ex sin 5x ?
Find the second order derivatives of the following function x3 log 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 x = a cos θ, y = b sin θ, show that \[\frac{d^2 y}{d x^2} = - \frac{b^4}{a^2 y^3}\] ?
If \[y = e^{\tan^{- 1} x}\] prove that (1 + x2)y2 + (2x − 1)y1 = 0 ?
If y = tan−1 x, show that \[\left( 1 + x^2 \right) \frac{d^2 y}{d x^2} + 2x\frac{dy}{dx} = 0\] ?
If y = cos−1 x, find \[\frac{d^2 y}{d x^2}\] in terms of y alone ?
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
\[\text { If y } = x^n \left\{ a \cos\left( \log x \right) + b \sin\left( \log x \right) \right\}, \text { prove that } x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0 \] Disclaimer: There is a misprint in the question. It must be
\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] instead of 1
\[x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0\] ?
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
If logy = tan–1 x, then show that `(1+x^2) (d^2y)/(dx^2) + (2x - 1) dy/dx = 0 .`
Find the minimum value of (ax + by), where xy = c2.
Differentiate sin(log sin x) ?
