Advertisements
Advertisements
Question
\[\frac{x + 1}{x + 2}\]
Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{x + h + 1}{x + h + 2} - \frac{x + 1}{x + 2}}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h + 1 \right)\left( x + 2 \right) - \left( x + h + 2 \right)\left( x + 1 \right)}{h\left( x + h + 2 \right)\left( x + 2 \right)}\]
\[ = \lim_{h \to 0} \frac{x^2 + 2x + hx + 2h + x + 2 - x^2 - x - hx - h - 2x - 2}{h\left( x + h + 2 \right)\left( x + 2 \right)}\]
\[ = \lim_{h \to 0} \frac{h}{h\left( x + h + 2 \right)\left( x + 2 \right)}\]
\[ = \lim_{h \to 0} \frac{1}{\left( x + h + 2 \right)\left( x + 2 \right)}\]
\[ = \frac{1}{\left( x + 0 + 2 \right)\left( x + 2 \right)}\]
\[ = \frac{1}{\left( x + 2 \right)^2}\]
APPEARS IN
RELATED QUESTIONS
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):
`(1 + 1/x)/(1- 1/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):
(ax + b)n (cx + d)m
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 f (x) = x2 − 2 at x = 10
\[\frac{1}{\sqrt{x}}\]
k xn
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
\[\sqrt{\tan x}\]
ex log a + ea long x + ea log a
(2x2 + 1) (3x + 2)
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
x3 ex
xn loga x
(x3 + x2 + 1) sin x
logx2 x
x4 (5 sin x − 3 cos x)
x5 (3 − 6x−9)
x−3 (5 + 3x)
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
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to
(ax2 + cot x)(p + q cos x)
