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 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):
(ax + b) (cx + d)2
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)/(cx + d)`
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 the following function at the indicated point:
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
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:
sin x + cos x
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
tan (2x + 1)
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
Find the rate at which the function f (x) = x4 − 2x3 + 3x2 + x + 5 changes with respect to x.
x3 ex
sin x cos x
sin2 x
logx2 x
x−3 (5 + 3x)
(ax + b) (a + d)2
\[\frac{e^x}{1 + x^2}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
