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 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):
`(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):
`(px^2 +qx + r)/(ax +b)`
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
\[\frac{x + 1}{x + 2}\]
(x2 + 1) (x − 5)
\[\sqrt{2 x^2 + 1}\]
x ex
Differentiate of the following from first principle:
sin (x + 1)
Differentiate of the following from first principle:
x sin x
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
tan 2x
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
xn tan x
\[\frac{2^x \cot x}{\sqrt{x}}\]
(1 − 2 tan x) (5 + 4 sin x)
logx2 x
x−3 (5 + 3x)
(ax + b)n (cx + d)n
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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 each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
(ax2 + cot x)(p + q cos x)
`(a + b sin x)/(c + d cos x)`
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
