Advertisements
Advertisements
प्रश्न
Differentiate \[\log \left( \tan^{- 1} x \right)\]?
Advertisements
उत्तर
\[\text{Let} y = \log\left( \tan^{- 1} x \right)\]
Differentiate it with respect to x we get,
\[\frac{d y}{d x} = \frac{d}{dx}\log\left( \tan^{- 1} x \right)\]
\[ = \frac{1}{\tan^{- 1} x} \times \frac{d}{dx}\left( \tan^{- 1} x \right) \left[ \text{Using chain rule} \right]\]
\[ = \frac{1}{\left( 1 + x^2 \right) \tan^{- 1} x}\]
\[So, \frac{d}{dx}\left\{ \log\left( \tan^{- 1} x \right) \right\} = \frac{1}{\left( 1 + x^2 \right) \tan^{- 1} x}\]
APPEARS IN
संबंधित प्रश्न
Differentiate the following functions from first principles sin−1 (2x + 3) ?
Differentiate sin (3x + 5) ?
Differentiate log7 (2x − 3) ?
Differentiate logx 3 ?
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
Differentiate \[\tan^{- 1} \left\{ \frac{x}{a + \sqrt{a^2 - x^2}} \right\}, - a < x < a\] ?
Differentiate \[\tan^{- 1} \left( \frac{x}{1 + 6 x^2} \right)\] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
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 \[\tan \left( x + y \right) + \tan \left( x - y \right) = 1, \text{ find} \frac{dy}{dx}\] ?
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 \[e^{\sin x }+ \left( \tan x \right)^x\] ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
If \[x = \left( t + \frac{1}{t} \right)^a , y = a^{t + \frac{1}{t}} , \text{ find } \frac{dy}{dx}\] ?
Write the derivative of sinx with respect to cos x ?
Differentiate \[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cos^{- 1} x, \text { if}\]\[x \in \left( 0, 1 \right)\] ?
Differentiate \[\left( \cos x \right)^{\sin x }\] with respect to \[\left( \sin x \right)^{\cos x }\]?
\[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1\] ?
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 \[y = x^x , \text{ find } \frac{dy}{dx} \text{ at } x = e\] ?
If \[x = a \cos^3 \theta, y = a \sin^3 \theta, \text { then } \sqrt{1 + \left( \frac{dy}{dx} \right)^2} =\] ____________ .
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
If \[y = \sqrt{\sin x + y}, \text { then }\frac{dy}{dx} \text { equals }\] ______________ .
If y = 500 e7x + 600 e−7x, show that \[\frac{d^2 y}{d x^2} = 49y\] ?
If y = (cot−1 x)2, prove that y2(x2 + 1)2 + 2x (x2 + 1) y1 = 2 ?
\[\text { If x } = a\left( \cos t + t \sin t \right) \text { and y} = a\left( \sin t - t \cos t \right),\text { then find the value of } \frac{d^2 y}{d x^2} \text { at } t = \frac{\pi}{4} \] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
\[\text { If y } = x^n \left\{ a \cos\left( \log x \right) + b \sin\left( \log x \right) \right\}, \text { prove that } x^2 \frac{d^2 y}{d x^2} + \left( 1 - 2n \right)x\frac{d y}{d x} + \left( 1 + n^2 \right)y = 0 \] 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 x = a cos nt − b sin nt and \[\frac{d^2 x}{dt} = \lambda x\] then find the value of λ ?
If x = t2 and y = t3, 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 x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
Find the minimum value of (ax + by), where xy = c2.
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Find the height of a cylinder, which is open at the top, having a given surface area, greatest volume, and radius r.
Range of 'a' for which x3 – 12x + [a] = 0 has exactly one real root is (–∞, p) ∪ [q, ∞), then ||p| – |q|| is ______.
