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
For the function
f(x) = `x^100/100 + x^99/99 + ...+ x^2/2 + x + 1`
Prove that f'(1) = 100 f'(0)
Find the derivative of `2x - 3/4`
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 f (x) = 99x at x = 100
\[\frac{2}{x}\]
\[\frac{x^2 - 1}{x}\]
\[\frac{1}{\sqrt{3 - x}}\]
x2 + x + 3
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan2 x
\[\sin \sqrt{2x}\]
\[\tan \sqrt{x}\]
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
x2 ex log x
xn loga x
x3 ex cos x
x4 (5 sin x − 3 cos x)
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
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\[f\left( x \right) = 1 + x + \frac{x^2}{2} + . . . + \frac{x^{100}}{100}\] then \[f'\left( 1 \right)\] is equal to
