Advertisements
Advertisements
प्रश्न
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Advertisements
उत्तर
\[\text{ Let y } = \left( \cos x \right)^x + \left( \sin x \right)^\frac{1}{x} \]
\[ \Rightarrow y = e^{ \log \left( \cos x\right)^x} + e^{\log \left( \sin x \right)^\frac{1}{x} } \]
\[ \Rightarrow y = e^{ x\log\left( \cos x \right) } + e^\frac{1}{x}\log\sin x\]
Differentiating with respect to x,
\[\frac{dy}{dx} = \frac{d}{dx}\left( e^{x \log\cos x} \right) + \frac{d}{dx}\left( e^\frac{1}{x}\log \sin x \right)\]
\[ = e^{x \log\cos x} \times \frac{d}{dx}\left( x \log\cos x \right) + e^\frac{1}{x}\log \sin x \frac{d}{dx}\left( \frac{1}{x}\log\sin x \right)\]
\[ = e^{\log \left( \cos x \right)^x }\times \left[ x\frac{d}{dx}\left( \log\cos x \right) + \log\cos x \times \frac{d}{dx}\left( x \right) \right] + e^{\log \left( \sin x \right)^\frac{1}{x} }\times \left[ \frac{1}{x}\frac{d}{dx}\left( \log\sin x \right) + \log\sin x\frac{d}{dx}\left( \frac{1}{x} \right) \right]\]
\[ = \left( \cos x \right)^x \left[ x\left( \frac{1}{\cos x} \right)\frac{d}{dx}\left( \cos x \right) + \log\cos x\left( 1 \right) \right] + \left( \sin \right)^\frac{1}{x} \left[ \frac{1}{x} \times \frac{1}{\sin x} \times \frac{d}{dx}\left( \sin x \right) + \log\sin x\left( - \frac{1}{x^2} \right) \right]\]
\[ = \left( \cos x \right)^x \left[ x\left( \frac{1}{\cos x} \right)\left( - \sin x \right) + \log\cos x \right] + \left( \sin x \right)^\frac{1}{x} \left[ \frac{1}{x} \times \frac{1}{\sin x}\left( \cos x \right) - \frac{1}{x^2}\log\sin x \right]\]
\[ = \left( \cos x \right)^x \left[ \log\cos x - x \tan x \right] + \left( \sin x \right)^\frac{1}{x} \left[ \frac{\cot x}{x} - \frac{1}{x^2}\log\sin x \right]\]
APPEARS IN
संबंधित प्रश्न
Prove that `y=(4sintheta)/(2+costheta)-theta `
Differentiate tan2 x ?
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{x} + \sqrt{a}}{1 - \sqrt{xa}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[\sin xy + \cos \left( x + y \right) = 1\] ?
If `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`
If \[x \sqrt{1 + y} + y \sqrt{1 + x} = 0\] , prove that \[\left( 1 + x \right)^2 \frac{dy}{dx} + 1 = 0\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
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 \[x = a e^\theta \left( \sin \theta - \cos \theta \right), y = a e^\theta \left( \sin \theta + \cos \theta \right)\] ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\] \[x \in \left( - 1, 0 \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
If f (x) = loge (loge x), then write the value of `f' (e)` ?
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 \[f\left( 1 \right) = 4, f'\left( 1 \right) = 2\] find the value of the derivative of \[\log \left( f\left( e^x \right) \right)\] w.r. to x at the point x = 0 ?
Let g (x) be the inverse of an invertible function f (x) which is derivable at x = 3. If f (3) = 9 and `f' (3) = 9`, write the value of `g' (9)`.
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
If \[\frac{\pi}{2} \leq x \leq \frac{3\pi}{2} \text { and y } = \sin^{- 1} \left( \sin x \right), \text { find } \frac{dy}{dx} \] ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[- \frac{\pi}{2} < x < 0 \text{ and y } = \tan^{- 1} \sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}, \text{ find } \frac{dy}{dx}\] ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .
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 ______________ .
If x = a (θ + sin θ), y = a (1 + cos θ), prove that \[\frac{d^2 y}{d x^2} = - \frac{a}{y^2}\] ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
\[\text { If x } = a\left( \cos2t + 2t \sin2t \right)\text { and y } = a\left( \sin2t - 2t \cos2t \right), \text { then find } \frac{d^2 y}{d x^2} \] ?
If \[y = 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!}\] .....to ∞, then write \[\frac{d^2 y}{d x^2}\] in terms of y ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
\[\frac{d^{20}}{d x^{20}} \left( 2 \cos x \cos 3 x \right) =\]
If y = a + bx2, a, b arbitrary constants, then
If x = f(t) and y = g(t), then \[\frac{d^2 y}{d x^2}\] is equal to
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
\[\text { If } y = \left( x + \sqrt{1 + x^2} \right)^n , \text { then show that }\]
\[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = n^2 y .\]
