Advertisements
Advertisements
Question
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 _____________ .
Options
1
0
\[x^{l + m + n}\]
none of these
Advertisements
Solution
`0`
We have,
\[ \Rightarrow f\left( x \right) = x^{l^2 - m^2} \times x^{m^2 - n^2} \times x^{n^2 - l^2} \]
\[ \Rightarrow f\left( x \right) = x^\left( l^2 - m^2 + m^2 - n^2 + n^2 - l^2 \right) \]
\[ \Rightarrow f\left( x \right) = x^0 \]
\[ \Rightarrow f\left( x \right) = 1\]
\[ \Rightarrow f'\left( x \right) = 0\]
APPEARS IN
RELATED QUESTIONS
If y = xx, prove that `(d^2y)/(dx^2)−1/y(dy/dx)^2−y/x=0.`
Differentiate the following functions from first principles \[e^\sqrt{2x}\].
Differentiate the following functions from first principles log cos x ?
Differentiate \[\tan^{- 1} \left( e^x \right)\] ?
Differentiate \[\frac{e^x \sin x}{\left( x^2 + 2 \right)^3}\] ?
If \[y = \frac{x \sin^{- 1} x}{\sqrt{1 - x^2}}\] , prove that \[\left( 1 - x^2 \right) \frac{dy}{dx} = x + \frac{y}{x}\] ?
If \[y = x \sin^{- 1} x + \sqrt{1 - x^2}\] ,prove that \[\frac{dy}{dx} = \sin^{- 1} x\] ?
Differentiate \[\cos^{- 1} \left( \frac{x + \sqrt{1 - x^2}}{\sqrt{2}} \right), - 1 < x < 1\] ?
Differentiate \[\sin^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right)\] ?
If \[y = \sin^{- 1} \left( \frac{x}{1 + x^2} \right) + \cos^{- 1} \left( \frac{1}{\sqrt{1 + x^2}} \right), 0 < x < \infty\] prove that \[\frac{dy}{dx} = \frac{2}{1 + x^2} \] ?
If the derivative of tan−1 (a + bx) takes the value 1 at x = 0, prove that 1 + a2 = b ?
If \[y = \sin^{- 1} \left( 6x\sqrt{1 - 9 x^2} \right), - \frac{1}{3\sqrt{2}} < x < \frac{1}{3\sqrt{2}}\] \[\frac{dy}{dx} \] ?
Differentiate \[{10}^\left( {10}^x \right)\] ?
Differentiate \[\left( \cos x \right)^x + \left( \sin x \right)^{1/x}\] ?
Find \[\frac{dy}{dx}\] \[y = e^x + {10}^x + x^x\] ?
Find \[\frac{dy}{dx}\] \[y = \left( \sin x \right)^{\cos x} + \left( \cos x \right)^{\sin x}\] ?
Find \[\frac{dy}{dx}\] \[y = x^{\log x }+ \left( \log x \right)^x\] ?
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)` ?
If \[y = \sqrt{x + \sqrt{x + \sqrt{x + . . . to \infty ,}}}\] prove that \[\frac{dy}{dx} = \frac{1}{2 y - 1}\] ?
If \[y = \sqrt{\log x + \sqrt{\log x + \sqrt{\log x + ... to \infty}}}\], prove that \[\left( 2 y - 1 \right) \frac{dy}{dx} = \frac{1}{x}\] ?
Find \[\frac{dy}{dx}\], when \[x = a t^2 \text{ and } y = 2\ at \] ?
Differentiate log (1 + x2) with respect to tan−1 x ?
Differentiate \[\sin^{- 1} \left( 4x \sqrt{1 - 4 x^2} \right)\] with respect to \[\sqrt{1 - 4 x^2}\] , if \[x \in \left( - \frac{1}{2}, - \frac{1}{2 \sqrt{2}} \right)\] ?
Differentiate \[\tan^{- 1} \left( \frac{1 + ax}{1 - ax} \right)\] with respect to \[\sqrt{1 + a^2 x^2}\] ?
If f (x) = loge (loge x), then write the value of `f' (e)` ?
If \[y = \log \left| 3x \right|, x \neq 0, \text{ find } \frac{dy}{dx} \] ?
The derivative of the function \[\cot^{- 1} \left| \left( \cos 2 x \right)^{1/2} \right| \text{ at } x = \pi/6 \text{ is }\] ______ .
If \[y = \sqrt{\sin x + y},\text { then } \frac{dy}{dx} =\] __________ .
If \[\sin y = x \sin \left( a + y \right), \text { then }\frac{dy}{dx} \text { is}\] ____________ .
If \[f\left( x \right) = \sqrt{x^2 - 10x + 25}\] then the derivative of f (x) in the interval [0, 7] is ____________ .
If \[y = \frac{1}{1 + x^{a - b} +^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}}\] then \[\frac{dy}{dx}\] is equal to ______________ .
If \[y = \log \sqrt{\tan x}\] then the value of \[\frac{dy}{dx}\text { at }x = \frac{\pi}{4}\] is given by __________ .
If y log (1 + cos x), prove that \[\frac{d^3 y}{d x^3} + \frac{d^2 y}{d x^2} \cdot \frac{dy}{dx} = 0\] ?
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 \[f\left( x \right) = \frac{\sin^{- 1} x}{\sqrt{1 - x^2}}\] then (1 − x)2 f '' (x) − xf(x) =
If y = (sin−1 x)2, then (1 − x2)y2 is equal to
If \[y = \frac{ax + b}{x^2 + c}\] then (2xy1 + y)y3 =
Differentiate `log [x+2+sqrt(x^2+4x+1)]`
