Advertisements
Advertisements
प्रश्न
\[\frac{x + 1}{x + 2}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{x + h + 1}{x + h + 2} - \frac{x + 1}{x + 2}}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h + 1 \right)\left( x + 2 \right) - \left( x + h + 2 \right)\left( x + 1 \right)}{h\left( x + h + 2 \right)\left( x + 2 \right)}\]
\[ = \lim_{h \to 0} \frac{x^2 + 2x + hx + 2h + x + 2 - x^2 - x - hx - h - 2x - 2}{h\left( x + h + 2 \right)\left( x + 2 \right)}\]
\[ = \lim_{h \to 0} \frac{h}{h\left( x + h + 2 \right)\left( x + 2 \right)}\]
\[ = \lim_{h \to 0} \frac{1}{\left( x + h + 2 \right)\left( x + 2 \right)}\]
\[ = \frac{1}{\left( x + 0 + 2 \right)\left( x + 2 \right)}\]
\[ = \frac{1}{\left( x + 2 \right)^2}\]
APPEARS IN
संबंधित प्रश्न
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):
sin (x + a)
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of f (x) = cos x at x = 0
Find the derivative of the following function at the indicated point:
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{x^2 - 1}{x}\]
(x2 + 1) (x − 5)
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
x sin x
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
(2x2 + 1) (3x + 2)
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
sin x cos x
(x sin x + cos x ) (ex + x2 log x)
x4 (5 sin x − 3 cos 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.
(x + 2) (x + 3)
(ax + b)n (cx + d)n
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{x^5 - \cos x}{\sin x}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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 \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
Find the derivative of f(x) = tan(ax + b), by first principle.
