Advertisements
Advertisements
Question
\[\frac{2}{x}\]
Advertisements
Solution
\[\left( i \right) \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{2}{x + h} - \frac{2}{x}}{h}\]
\[ = \lim_{h \to 0} \frac{2x - 2x - 2h}{hx(x + h)}\]
\[ = \lim_{h \to 0} \frac{- 2h}{hx(x + h)}\]
\[ = \lim_{h \to 0} \frac{- 2}{x(x + h)}\]
\[ = \frac{- 2}{x^2}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of 99x at x = 100.
Find the derivative of x–3 (5 + 3x).
Find the derivative of x5 (3 – 6x–9).
Find the derivative of x–4 (3 – 4x–5).
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)`
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):
cosec x cot x
\[\frac{x^2 + 1}{x}\]
k xn
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
ex log a + ea long x + ea log a
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
(x3 + x2 + 1) sin x
sin x cos x
\[\frac{2^x \cot x}{\sqrt{x}}\]
(x sin x + cos x ) (ex + x2 log 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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{x}{1 + \tan x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
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 f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
