Advertisements
Advertisements
प्रश्न
Differentiate \[\frac{x^2 \left( 1 - x^2 \right)}{\cos 2x}\] ?
उत्तर
\[\text{Let } y = \frac{x^2 \left( 1 - x^2 \right)^3}{\cos 2x}\]
\[\Rightarrow \frac{d y}{d x} = \frac{\cos2x\frac{d}{dx}\left\{ x^2 \left( 1 - x^2 \right)^3 \right\} - x^2 \left( 1 - x^2 \right)^3 \frac{d}{dx}\cos2x}{\cos^2 2x} \]
\[ = \frac{\cos2x\left\{ x^2 \frac{d}{dx} \left( 1 - x^2 \right)^3 + \left( 1 - x^2 \right)^3 \frac{d}{dx}\left( x^2 \right) \right\} - x^2 \left( 1 - x^2 \right)^3 \left( - 2\sin2x \right)}{\cos^2 2x}\]
\[ = \frac{\cos2x\left\{ - 6 x^3 \left( 1 - x^2 \right)^2 + \left( 1 - x^2 \right)^3 2x \right\} + 2 x^2 \left( 1 - x^2 \right)^3 \sin2x}{\cos^2 2x}\]
\[ = \frac{2x \left( 1 - x^2 \right)^2}{\cos2x} - \frac{6 x^3 \left( 1 - x^2 \right)^2}{\cos2x} + \frac{2 x^2 \left( 1 - x^2 \right)^3 \sin2x}{\cos^2 2x}\]
\[ = 2x\left( 1 - x^2 \right)\sec2x\left\{ 1 - 4 x^2 + x\left( 1 - x^2 \right)\tan2x \right\}\]
\[So, \frac{d}{dx}\left\{ \frac{x^2 \left( 1 - x^2 \right)^3}{\cos2x} \right\} = 2x\left( 1 - x^2 \right)\sec2x\left\{ 1 - 4 x^2 + x\left( 1 - x^2 \right)\tan2x \right\}\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles e−x.
Differentiate \[\sqrt{\frac{1 - x^2}{1 + x^2}}\] ?
Differentiate \[\log \sqrt{\frac{1 - \cos x}{1 + \cos x}}\] ?
Differentiate \[\log \left( cosec x - \cot x \right)\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
If \[y = \sqrt{x + 1} + \sqrt{x - 1}\] , prove that \[\sqrt{x^2 - 1}\frac{dy}{dx} = \frac{1}{2}y\] ?
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\], prove that \[2 x\frac{dy}{dx} = \sqrt{x} - \frac{1}{\sqrt{x}}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\tan^{- 1} \left( \frac{2 a^x}{1 - a^{2x}} \right), a > 1, - \infty < x < 0\] ?
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 \[\frac{dy}{dx} = \frac{4}{1 + x^2} \] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] ?
Find \[\frac{dy}{dx}\] in the following case \[\tan^{- 1} \left( x^2 + y^2 \right) = a\] ?
If \[\sqrt{1 - x^2} + \sqrt{1 - y^2} = a \left( x - y \right)\] , prove that \[\frac{dy}{dx} = \frac{\sqrt{1 - y^2}}{1 - x^2}\] ?
If \[\sin \left( xy \right) + \frac{y}{x} = x^2 - y^2 , \text{ find} \frac{dy}{dx}\] ?
If \[\cos y = x \cos \left( a + y \right), \text{ with } \cos a \neq \pm 1, \text{ prove that } \frac{dy}{dx} = \frac{\cos^2 \left( a + y \right)}{\sin a}\] ?
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\cos x} + \left( \sin x \right)^{\tan x}\] ?
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 \[x = \frac{e^t + e^{- t}}{2} \text{ and } y = \frac{e^t - e^{- t}}{2}\] ?
Find \[\frac{dy}{dx}\] ,When \[x = e^\theta \left( \theta + \frac{1}{\theta} \right) \text{ and } y = e^{- \theta} \left( \theta - \frac{1}{\theta} \right)\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
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 ?
If \[y = \sin^{- 1} \left( \frac{2x}{1 + x^2} \right)\] write the value of \[\frac{dy}{dx}\text { for } x > 1\] ?
If \[x = 3\sin t - \sin3t, y = 3\cos t - \cos3t \text{ find }\frac{dy}{dx} \text{ at } t = \frac{\pi}{3}\] ?
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
Find the second order derivatives of the following function sin (log x) ?
Find the second order derivatives of the following function x3 log x ?
If y = 2 sin x + 3 cos x, show that \[\frac{d^2 y}{d x^2} + y = 0\] ?
If \[y = \left[ \log \left( x + \sqrt{x^2 + 1} \right) \right]^2\] show that \[\left( 1 + x^2 \right)\frac{d^2 y}{d x^2} + x\frac{dy}{dx} = 2\] ?
If y = (tan−1 x)2, then prove that (1 + x2)2 y2 + 2x(1 + x2)y1 = 2 ?
If y = cot x show that \[\frac{d^2 y}{d x^2} + 2y\frac{dy}{dx} = 0\] ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
If y = a + bx2, a, b arbitrary constants, then
If y = a sin mx + b cos mx, then \[\frac{d^2 y}{d x^2}\] is equal to
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
Differentiate \[\tan^{- 1} \left( \frac{\sqrt{1 + x^2} - 1}{x} \right) w . r . t . \sin^{- 1} \frac{2x}{1 + x^2},\]tan-11+x2-1x w.r.t. sin-12x1+x2, if x ∈ (–1, 1) .
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
