Advertisements
Advertisements
Question
\[\frac{2x + 3}{x - 2}\]
Advertisements
Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{2\left( x + h \right) + 3}{x + h - 2} - \frac{2x + 3}{x - 2}}{h}\]
\[ = \lim_{h \to 0} \frac{\left( 2x + 2h + 3 \right)\left( x - 2 \right) - \left( x + h - 2 \right)\left( 2x + 3 \right)}{h\left( x + h - 2 \right)\left( x - 2 \right)}\]
\[ = \lim_{h \to 0} \frac{2 x^2 + 2xh + 3x - 4x - 4h - 6 - 2 x^2 - 2xh + 4x - 3x - 3h + 6}{h\left( x + h - 2 \right)\left( x - 2 \right)}\]
\[ = \lim_{h \to 0} \frac{- 7h}{h\left( x + h - 2 \right)\left( x - 2 \right)}\]
\[ = \lim_{h \to 0} \frac{- 7}{\left( x + h - 2 \right)\left( x - 2 \right)}\]
\[ = \frac{- 7}{\left( x - 2 \right)\left( x - 2 \right)}\]
\[ = \frac{- 7}{\left( x - 2 \right)^2}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of (5x3 + 3x – 1) (x – 1).
Find the derivative of x5 (3 – 6x–9).
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
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)`
\[\frac{2}{x}\]
\[\frac{1}{\sqrt{x}}\]
\[\frac{x + 2}{3x + 5}\]
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
tan 2x
3x + x3 + 33
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
(x sin x + cos x) (x cos x − sin x)
sin2 x
x5 (3 − 6x−9)
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{x + \cos x}{\tan x}\]
\[\frac{x}{\sin^n x}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
Mark the correct alternative in of the following:
If\[y = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + . . .\]then \[\frac{dy}{dx} =\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
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
