Advertisements
Advertisements
प्रश्न
\[\frac{1}{x^3}\]
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{1}{(x + h )^3} - \frac{1}{x^3}}{h}\]
\[ = \lim_{h \to 0} \frac{x^3 - (x + h )^3}{h(x + h )^3 x^3}\]
\[ = \lim_{h \to 0} \frac{x^3 - x^3 - 3 x^2 h - 3x h^2 - h^3}{h(x + h )^3 x^3}\]
\[ = \lim_{h \to 0} \frac{- 3 x^2 h - 3x h^2 - h^3}{h(x + h )^3 x^3}\]
\[ = \lim_{h \to 0} \frac{h\left( - 3 x^2 - 3xh - h^2 \right)}{h(x + h )^3 x^3}\]
\[ = \lim_{h \to 0} \frac{\left( - 3 x^2 - 3xh - h^2 \right)}{(x + h )^3 x^3}\]
\[ = \frac{- 3 x^2}{x^6}\]
\[ = \frac{- 3}{x^4}\]
\[ = - 3 x^{- 4} \]
\[\]
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 `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):
(ax + b)n (cx + d)m
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 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))/ cos x`
Find the derivative of f (x) = 3x at x = 2
\[\frac{1}{\sqrt{x}}\]
k xn
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
e3x
Differentiate of the following from first principle:
eax + b
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
tan2 x
tan 2x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
cos (x + a)
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
(1 − 2 tan x) (5 + 4 sin x)
logx2 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
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{1 + \log x}{1 - \log x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{x + \cos x}{\tan x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
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 \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
