Advertisements
Advertisements
Question
Differentiate \[\tan^{- 1} \left\{ \frac{x^{1/3} + a^{1/3}}{1 - \left( a x \right)^{1/3}} \right\}\] ?
Advertisements
Solution
\[\text{ Let, y } = \tan^{- 1} \left[ \frac{x^\frac{1}{3} + a^\frac{1}{3}}{1 - \left( ax \right)^\frac{1}{3}} \right]\]
\[ \Rightarrow y = \tan^{- 1} \left( x^\frac{1}{3} \right) + \tan^{- 1} \left( a^\frac{1}{3} \right) \left[ \text{ Since }, \tan^{- 1} x + \tan^{- 1} y = \tan^{- 1} \left( \frac{x + y}{1 - xy} \right) \right]\]
Differentiate it with respect to x using chain rule,
\[\frac{d y}{d x} = \frac{1}{1 + \left( x^\frac{1}{3} \right)^2} \times \frac{d}{dx}\left( x^\frac{1}{3} \right) + 0\]
\[ \Rightarrow \frac{d y}{d x} = \frac{\left( \frac{1}{3} \times x^{\frac{1}{3} - 1} \right)}{1 + x^\frac{2}{3}}\]
\[ \therefore \frac{d y}{d x} = \frac{1}{3 x^\frac{2}{3} \left( 1 + x^\frac{2}{3} \right)}\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles e3x.
Differentiate sin (log x) ?
Differentiate \[e^{\sin} \sqrt{x}\] ?
Differentiate \[3^{e^x}\] ?
Differentiate \[\frac{e^{2x} + e^{- 2x}}{e^{2x} - e^{- 2x}}\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
Differentiate \[\log \sqrt{\frac{x - 1}{x + 1}}\] ?
If \[y = \left( x - 1 \right) \log \left( x - 1 \right) - \left( x + 1 \right) \log \left( x + 1 \right)\] , prove that \[\frac{dy}{dc} = \log \left( \frac{x - 1}{1 + x} \right)\] ?
If \[y = \sqrt{x^2 + a^2}\] prove that \[y\frac{dy}{dx} - x = 0\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{x}{\sqrt{x^2 + a^2}} \right\}\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\sin^{- 1} \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{2} \right\}, 0 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{\sin x}{1 + \cos x} \right), - \pi < x < \pi\] ?
Differentiate the following with respect to x:
\[\cos^{- 1} \left( \sin x \right)\]
If \[y = \tan^{- 1} \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt{1 + x} + \sqrt{1 - x}} \right), \text{find } \frac{dy}{dx}\] ?
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
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 \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
Find \[\frac{dy}{dx}\] \[y = \sin x \sin 2x \sin 3x \sin 4x\] ?
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 \[\frac{dy}{dx}\] , when \[x = \frac{1 - t^2}{1 + t^2} \text{ and y } = \frac{2 t}{1 + t^2}\] ?
Differentiate \[\tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right)\] with respect to \[\sec^{- 1} x\] ?
If \[f\left( x \right) = x + 1\] , then write the value of \[\frac{d}{dx} \left( fof \right) \left( x \right)\] ?
If \[y = \sec^{- 1} \left( \frac{x + 1}{x - 1} \right) + \sin^{- 1} \left( \frac{x - 1}{x + 1} \right)\] then write the value of \[\frac{dy}{dx} \] ?
Differential coefficient of sec(tan−1 x) is ______.
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If \[y = \log \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] __________ .
If y = log (sin x), prove that \[\frac{d^3 y}{d x^3} = 2 \cos \ x \ {cosec}^3 x\] ?
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 = (sin−1 x)2, prove that (1 − x2)
\[\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] ?
If \[y = \left| \log_e x \right|\] find\[\frac{d^2 y}{d x^2}\] ?
If x = t2, y = t3, then \[\frac{d^2 y}{d x^2} =\]
If \[y = \log_e \left( \frac{x}{a + bx} \right)^x\] then x3 y2 =
If x = sin t and y = sin pt, prove that \[\left( 1 - x^2 \right)\frac{d^2 y}{d x^2} - x\frac{dy}{dx} + p^2 y = 0\] .
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
f(x) = xx has a stationary point at ______.
