Advertisements
Advertisements
Question
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
Advertisements
Solution
\[\frac{d}{dx} \left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3 \]
\[ = \frac{d}{dx}\left[ \left( \sqrt{x} \right)^3 + 3 \left( \sqrt{x} \right)^2 \left( \frac{1}{\sqrt{x}} \right) + 3\left( \sqrt{x} \right) \left( \frac{1}{\sqrt{x}} \right)^2 + \left( \frac{1}{\sqrt{x}} \right)^3 \right]\]
\[ = \frac{d}{dx}\left( x^\frac{3}{2} \right) + 3\frac{d}{dx}\left( x^\frac{1}{2} \right) + 3\frac{d}{dx}\left( x^\frac{- 1}{2} \right) + \frac{d}{dx}\left( x^\frac{- 3}{2} \right)\]
\[ = \frac{3}{2} x^\frac{3}{2} - 1 + 3 . \frac{1}{2} x^\frac{1}{2} - 1 + 3\left( \frac{- 1}{2} \right) x^\frac{- 1}{2} - 1 + \left( \frac{- 3}{2} \right) x^\frac{- 3}{2} - 1 \]
\[ = \frac{3}{2} x^\frac{1}{2} + \frac{3}{2} x^\frac{- 1}{2} - \frac{3}{2} x^\frac{- 3}{2} - \frac{3}{2} x^\frac{- 5}{2}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of `2x - 3/4`
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`(px+ q) (r/s + s)`
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
(ax + b) (cx + d)2
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`1/(ax^2 + bx + c)`
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`cos x/(1 + sin x)`
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
`(a + bsin x)/(c + dcosx)`
Find the derivative of the following function (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
x4 (5 sin x – 3 cos x)
Find the derivative of the following function at the indicated point:
\[\frac{x^2 + 1}{x}\]
(x2 + 1) (x − 5)
x ex
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
x cos x
Differentiate of the following from first principle:
sin (2x − 3)
tan2 x
tan (2x + 1)
(2x2 + 1) (3x + 2)
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
x3 ex
x5 ex + x6 log x
(1 − 2 tan x) (5 + 4 sin x)
logx2 x
\[\frac{e^x}{1 + x^2}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Find the derivative of 2x4 + x.
`(a + b sin x)/(c + d cos x)`
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
