Advertisements
Advertisements
Question
Differentiate \[\frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\] ?
Solution
\[\text{Let } y = \frac{3 x^2 \sin x}{\sqrt{7 - x^2}}\]
Differentiate it with respect to x we get,
\[\frac{d y}{d x} = \frac{d}{dx}\left\{ \frac{3 x^2 sinx}{\left( 7 - x^2 \right)^\frac{1}{2}} \right\}\]
\[ = \frac{\left( 7 - x^2 \right)^\frac{1}{2} \times \frac{d}{dx}\left( 3 x^2 \sin x \right) - \left( 3 x^2 \sin x \right)\frac{d}{dx} \left( 7 - x^2 \right)^\frac{1}{2}}{\left[ \left( 7 - x^2 \right)^\frac{1}{2} \right]^2} \left[ \text{Using quotient rule, chain rule and product rule} \right]\]
\[ = \left[ \frac{\left( 7 - x^2 \right)^\frac{1}{2} \times 3\left[ x^2 \frac{d}{dx}\left( \sin x \right) + \sin x\frac{d}{dx}\left( x^2 \right) \right] - 3 x^2 \sin x \times \frac{1}{2}\left( 7 - x^2 \right) \times \frac{d}{dx}\left( 7 - x^2 \right)}{\left( 7 - x^2 \right)} \right]\]
\[ = \left[ \frac{\left( 7 - x^2 \right)^\frac{1}{2} 3\left( x^2 \cos x + 2x \sin x \right) - 3 x^2 \sin x \times \frac{1}{2} \left( 7 - x^2 \right)^\frac{- 1}{2} \left( - 2x \right)}{\left( 7 - x^2 \right)} \right]\]
\[ = \left[ \frac{\left( 7 - x^2 \right)^\frac{1}{2} \times 3\left( x^2 \cos x + 2x \sin x \right)}{\left( 7 - x^2 \right)} + \frac{3 x^3 \sin x \left( 7 - x^2 \right)^\frac{- 1}{2}}{\left( 7 - x^2 \right)} \right]\]
\[ = \left[ \frac{6x \sin x + 3 x^2 \cos x}{\sqrt{\left( 7 - x^2 \right)}} + \frac{3 x^3 \sin x}{\left( 7 - x^2 \right)^\frac{3}{2}} \right]\]
\[So, \frac{d}{dx}\left( \frac{3 x^2 \sin x}{\sqrt{7 - x^2}} \right) = \left[ \frac{6x \sin x + 3 x^2 \cos x}{\sqrt{\left( 7 - x^2 \right)}} + \frac{3 x^3 \sin x}{\left( 7 - x^2 \right)^\frac{3}{2}} \right]\]
APPEARS IN
RELATED QUESTIONS
Differentiate the following functions from first principles ecos x.
Differentiate \[\sin \left( \frac{1 + x^2}{1 - x^2} \right)\] ?
Differentiate \[\frac{2^x \cos x}{\left( x^2 + 3 \right)^2}\] ?
Differentiate \[\left( \sin^{- 1} x^4 \right)^4\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
If \[y = \log \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]prove that \[\frac{dy}{dx} = \frac{x - 1}{2x \left( x + 1 \right)}\] ?
If xy = 4, prove that \[x\left( \frac{dy}{dx} + y^2 \right) = 3 y\] ?
Differentiate \[\tan^{- 1} \left( \frac{a + b \tan x}{b - a \tan x} \right)\] ?
If \[y = x \sin y\] , Prove that \[\frac{dy}{dx} = \frac{\sin y}{\left( 1 - x \cos y \right)}\] ?
Differentiate \[\left( \sin x \right)^{\log x}\] ?
Differentiate \[\sin \left( x^x \right)\] ?
Differentiate \[x^{x \cos x +} \frac{x^2 + 1}{x^2 - 1}\] ?
Differentiate\[\left( x + \frac{1}{x} \right)^x + x^\left( 1 + \frac{1}{x} \right)\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \tan x \right)^{\log x} + \cos^2 \left( \frac{\pi}{4} \right)\] ?
If \[x^x + y^x = 1\], prove that \[\frac{dy}{dx} = - \left\{ \frac{x^x \left( 1 + \log x \right) + y^x \cdot \log y}{x \cdot y^\left( x - 1 \right)} \right\}\] ?
If \[\left( \sin x \right)^y = x + y\] , prove that \[\frac{dy}{dx} = \frac{1 - \left( x + y \right) y \cot x}{\left( x + y \right) \log \sin x - 1}\] ?
If \[y = x \sin y\] , prove that \[\frac{dy}{dx} = \frac{y}{x \left( 1 - x \cos y \right)}\] ?
Find \[\frac{dy}{dx}\] , when \[x = \frac{3 at}{1 + t^2}, \text{ and } y = \frac{3 a t^2}{1 + t^2}\] ?
Differentiate x2 with respect to x3
\[\sin^{- 1} \sqrt{1 - x^2}\] with respect to \[\cot^{- 1} \left( \frac{x}{\sqrt{1 - x^2}} \right),\text { if }0 < x < 1\] ?
If \[y = \sin^{- 1} \left( \sin x \right), - \frac{\pi}{2} \leq x \leq \frac{\pi}{2}\] ,Then, write the value of \[\frac{dy}{dx} \text{ for } x \in \left( - \frac{\pi}{2}, \frac{\pi}{2} \right) \] ?
If \[y = \log \sqrt{\tan x}, \text{ write } \frac{dy}{dx} \] ?
If \[f\left( x \right) = \log \left\{ \frac{u \left( x \right)}{v \left( x \right)} \right\}, u \left( 1 \right) = v \left( 1 \right) \text{ and }u' \left( 1 \right) = v' \left( 1 \right) = 2\] , then find the value of `f' (1)` ?
Differential coefficient of sec(tan−1 x) is ______.
If \[y = \sin^{- 1} \left( \frac{1 - x^2}{1 + x^2} \right), \text { then } \frac{dy}{dx} =\] _____________ .
\[\frac{d}{dx} \left\{ \tan^{- 1} \left( \frac{\cos x}{1 + \sin x} \right) \right\} \text { equals }\] ______________ .
Find the second order derivatives of the following function e6x cos 3x ?
Find the second order derivatives of the following function tan−1 x ?
If y = ex cos x, prove that \[\frac{d^2 y}{d x^2} = 2 e^x \cos \left( x + \frac{\pi}{2} \right)\] ?
If log y = tan−1 x, show that (1 + x2)y2 + (2x − 1) y1 = 0 ?
Find \[\frac{d^2 y}{d x^2}\] where \[y = \log \left( \frac{x^2}{e^2} \right)\] ?
If x = 4z2 + 5, y = 6z2 + 7z + 3, find \[\frac{d^2 y}{d x^2}\] ?
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
If y = cosec−1 x, x >1, then show that \[x\left( x^2 - 1 \right)\frac{d^2 y}{d x^2} + \left( 2 x^2 - 1 \right)\frac{dy}{dx} = 0\] ?
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 = axn+1 + bx−n, then \[x^2 \frac{d^2 y}{d x^2} =\]
If \[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\] then (1 − x)2 f '' (x) − xf(x) =
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
If `x=a (cos t +t sint )and y= a(sint-cos t )` Prove that `Sec^3 t/(at),0<t< pi/2`
