Advertisements
Advertisements
Question
\[\frac{x + 2}{3x + 5}\]
Advertisements
Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{x + h + 2}{3\left( x + h \right) + 5} - \frac{x + 2}{3x + 5}}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{x + h + 2}{3x + 3h + 5} - \frac{x + 2}{3x + 5}}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h + 2 \right)\left( 3x + 5 \right) - \left( 3x + 3h + 5 \right)\left( x + 2 \right)}{h\left( 3x + 3h + 5 \right)\left( 3x + 5 \right)}\]
\[ = \lim_{h \to 0} \frac{3 x^2 + 3xh + 6x + 5x + 5h + 10 - 3 x^2 - 3xh - 5x - 6x - 6h - 10}{h\left( 3x + 3h + 5 \right)\left( 3x + 5 \right)}\]
\[ = \lim_{h \to 0} \frac{- h}{h\left( 3x + 3h + 5 \right)\left( 3x + 5 \right)}\]
\[ = \lim_{h \to 0} \frac{- 1}{\left( 3x + 3h + 5 \right)\left( 3x + 5 \right)}\]
\[ = \frac{- 1}{\left( 3x + 5 \right)^2}\]
APPEARS IN
RELATED QUESTIONS
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):
(x + a)
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):
`(ax + b)/(px^2 + qx + r)`
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 at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 + 1}{x}\]
x2 + x + 3
\[\frac{2x + 3}{x - 2}\]
x ex
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
tan (2x + 1)
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
3x + x3 + 33
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
x3 sin x
x2 ex log x
(1 − 2 tan x) (5 + 4 sin 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.
(x + 2) (x + 3)
(ax + b) (a + d)2
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{x}{\sin^n x}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
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\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to
Mark the correct alternative in of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
(ax2 + cot x)(p + q cos x)
