Advertisements
Advertisements
Question
\[\frac{1}{\sqrt{x}}\]
Advertisements
Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{1}{\sqrt{x + h}} - \frac{1}{\sqrt{x}}}{h}\]
\[ = \lim_{h \to 0} \frac{\sqrt{x} - \sqrt{x + h}}{h\sqrt{x}\sqrt{x + h}} \times \frac{\sqrt{x} + \sqrt{x + h}}{\sqrt{x} + \sqrt{x + h}}\]
\[ = \lim_{h \to 0} \frac{x - x - h}{h\sqrt{x}\sqrt{x + h}\left( \sqrt{x} + \sqrt{x + h} \right)}\]
\[ = \lim_{h \to 0} \frac{- h}{h\sqrt{x}\sqrt{x + h}\left( \sqrt{x} + \sqrt{x + h} \right)}\]
\[ = \lim_{h \to 0} \frac{- 1}{\sqrt{x}\sqrt{x + h}\left( \sqrt{x} + \sqrt{x + h} \right)}\]
\[ = \frac{- 1}{\sqrt{x}\sqrt{x}\left( \sqrt{x} + \sqrt{x} \right)}\]
\[ = \frac{- 1}{x \times 2\sqrt{x}}\]
\[ = \frac{- 1}{2 x^\frac{3}{2}}\]
\[ = - \frac{1}{2} x^\frac{- 3}{2} \]
\[\]
APPEARS IN
RELATED QUESTIONS
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
Find the derivative of x5 (3 – 6x–9).
Find the derivative of `2/(x + 1) - x^2/(3x -1)`.
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):
`a/x^4 = b/x^2 + cos 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):
`4sqrtx - 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):
cosec x cot 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):
`(sin(x + a))/ cos x`
Find the derivative of f (x) = tan x at x = 0
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 - 1}{x}\]
\[\frac{1}{\sqrt{3 - x}}\]
x2 + x + 3
(x2 + 1) (x − 5)
\[\sqrt{2 x^2 + 1}\]
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
x2 ex
tan 2x
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x3 ex
x2 sin x log x
sin2 x
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3x2 + 2)2
Differentiate each of the following functions by the product rule and the other method and verify that answer from both the methods is the same.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
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 \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
Find the derivative of f(x) = tan(ax + b), by first principle.
(ax2 + cot x)(p + q cos x)
