Advertisements
Advertisements
प्रश्न
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
Advertisements
उत्तर
We have, \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\]
\[So, \frac{dy}{dx} = \frac{d}{dx}\left[ \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right) \right]\]
\[ = \frac{d}{dx}\left[ \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right) \right]\]
\[ = \frac{1}{\sqrt{1 - \left( 6x\sqrt{1 - 9 x^2} \right)^2}} \times \frac{d}{dx}\left( 6x\sqrt{1 - 9 x^2} \right)\]
\[ = \frac{1}{\sqrt{1 - \left[ 36 x^2 \left( 1 - 9 x^2 \right) \right]}} \times \left( 6x\frac{d}{dx}\sqrt{1 - 9 x^2} + \sqrt{1 - 9 x^2}\frac{d}{dx}\left( 6x \right) \right)\]
\[ = \frac{1}{\sqrt{1 - 36 x^2 - 324 x^4}} \times \left( 6x \times \frac{1}{2\sqrt{1 - 9 x^2}}\frac{d}{dx}\left( 1 - 9 x^2 \right) + \sqrt{1 - 9 x^2}\left( 6 \right) \right)\]
\[ = \frac{1}{\sqrt{1 - 36 x^2 - 324 x^4}} \times \left( 6x \times \frac{1}{2\sqrt{1 - 9 x^2}} \times \left( - 18x \right) + 6\sqrt{1 - 9 x^2} \right)\]
\[ = \frac{1}{\sqrt{1 - 36 x^2 - 324 x^4}} \times \left( \frac{- 54 x^2}{\sqrt{1 - 9 x^2}} + 6\sqrt{1 - 9 x^2} \right)\]
\[ = \frac{1}{\sqrt{1 - 36 x^2 - 324 x^4}} \times \left( \frac{- 54 x^2 + 6\left( 1 - 9 x^2 \right)}{\sqrt{1 - 9 x^2}} \right)\]
\[ = \frac{- 54 x^2 + 6 - 54 x^2}{\sqrt{1 - 9 x^2}\sqrt{1 - 36 x^2 - 324 x^4}}\]
\[ = \frac{6 - 108 x^2}{\sqrt{1 - 9 x^2}\sqrt{1 - 36 x^2 - 324 x^4}}\]
APPEARS IN
संबंधित प्रश्न
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Differentiate sin (3x + 5) ?
Differentiate \[\sqrt{\frac{1 + \sin x}{1 - \sin x}}\] ?
Differentiate \[e^{\tan 3 x} \] ?
Differentiate \[\log \left( cosec x - \cot x \right)\] ?
Differentiate \[\frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}}\] ?
If \[y = \frac{x}{x + 2}\] , prove tha \[x\frac{dy}{dx} = \left( 1 - y \right) y\] ?
Differentiate \[\sin^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + bx}{b - ax} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{5 x}{1 - 6 x^2} \right), - \frac{1}{\sqrt{6}} < x < \frac{1}{\sqrt{6}}\] ?
Find \[\frac{dy}{dx}\] in the following case \[4x + 3y = \log \left( 4x - 3y \right)\] ?
If `ysqrt(1-x^2) + xsqrt(1-y^2) = 1` prove that `dy/dx = -sqrt((1-y^2)/(1-x^2))`
Differentiate \[\left( \log x \right)^{\cos x}\] ?
Differentiate \[x^\left( \sin x - \cos x \right) + \frac{x^2 - 1}{x^2 + 1}\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
find \[\frac{dy}{dx}\] \[y = \frac{\left( x^2 - 1 \right)^3 \left( 2x - 1 \right)}{\sqrt{\left( x - 3 \right) \left( 4x - 1 \right)}}\] ?
If \[y = \sqrt{\tan x + \sqrt{\tan x + \sqrt{\tan x + . . to \infty}}}\] , prove that \[\frac{dy}{dx} = \frac{\sec^2 x}{2 y - 1}\] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate (log x)x with respect to log x ?
Differentiate \[\sin^{- 1} \left( 2x \sqrt{1 - x^2} \right)\] with respect to \[\sec^{- 1} \left( \frac{1}{\sqrt{1 - x^2}} \right)\], if \[x \in \left( \frac{1}{\sqrt{2}}, 1 \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{2x}{1 - x^2} \right)\] with respect to \[\cos^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right),\text { if }0 < x < 1\] ?
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 \[\frac{dy}{dx} \] ?
If f (x) is an even function, then write whether `f' (x)` is even or odd ?
If \[x^y = e^{x - y} ,\text{ then } \frac{dy}{dx}\] is __________ .
If \[\sin \left( x + y \right) = \log \left( x + y \right), \text { then } \frac{dy}{dx} =\] ___________ .
If \[3 \sin \left( xy \right) + 4 \cos \left( xy \right) = 5, \text { then } \frac{dy}{dx} =\] _____________ .
If \[f\left( x \right) = \left( \frac{x^l}{x^m} \right)^{l + m} \left( \frac{x^m}{x^n} \right)^{m + n} \left( \frac{x^n}{x^l} \right)^{n + 1}\] the f' (x) is equal to _____________ .
Find the second order derivatives of the following function ex sin 5x ?
If y = sin (log x), prove that \[x^2 \frac{d^2 y}{d x^2} + x\frac{dy}{dx} + y = 0\] ?
\[\text { If x } = a \sin t - b \cos t, y = a \cos t + b \sin t, \text { prove that } \frac{d^2 y}{d x^2} = - \frac{x^2 + y^2}{y^3} \] ?
\[\text { Find A and B so that y = A } \sin3x + B \cos3x \text { satisfies the equation }\]
\[\frac{d^2 y}{d x^2} + 4\frac{d y}{d x} + 3y = 10 \cos3x \] ?
If x = f(t) and y = g(t), then write the value of \[\frac{d^2 y}{d x^2}\] ?
f(x) = xx has a stationary point at ______.
f(x) = 3x2 + 6x + 8, x ∈ R
