Advertisements
Advertisements
प्रश्न
\[\frac{x^2 + 1}{x}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{(x + h )^2 + 1}{x + h} - \frac{x^2 + 1}{x}}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{x^2 + 2xh + h^2 + 1}{x + h} - \frac{x^2 + 1}{x}}{h}\]
\[ = \lim_{h \to 0} \frac{x^3 + 2 x^2 h + h^2 x + x - x^3 - x^2 h - x - h}{xh(x + h)}\]
\[ = \lim_{h \to 0} \frac{x^2 h + h^2 x - h}{x(x + h)}\]
\[ = \lim_{h \to 0} \frac{h( x^2 + hx - 1)}{xh(x + h)}\]
\[ = \lim_{h \to 0} \frac{x^2 + hx - 1}{x(x + h)}\]
\[ = \frac{x^2 - 1}{x^2}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
Find the derivative of 99x at x = 100.
Find the derivative of (5x3 + 3x – 1) (x – 1).
Find the derivative of x–3 (5 + 3x).
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):
cosec x cot x
Find the derivative of f (x) = tan x at x = 0
\[\frac{2}{x}\]
\[\frac{1}{\sqrt{x}}\]
\[\sqrt{2 x^2 + 1}\]
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
eax + b
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan2 x
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
x2 ex log x
x2 sin x log x
(x sin x + cos x ) (ex + x2 log x)
(2x2 − 3) sin x
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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Write the derivative of f (x) = 3 |2 + x| at x = −3.
Find the derivative of f(x) = tan(ax + b), by first principle.
