Advertisements
Advertisements
प्रश्न
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\frac{\cos \left( x + h \right)}{x + h} - \frac{\cos x}{x}}{h}\]
\[ = \lim_{h \to 0} \frac{x \cos \left( x + h \right) - \left( x + h \right) \cos x}{h x \left( x + h \right)}\]
\[ = \lim_{h \to 0} \frac{x \left( \cos x \cos h - \sin x \sin h \right) - x \cos x - h \cos x}{h x \left( x + h \right)}\]
\[ = \lim_{h \to 0} \frac{x \cos x \cos h - x \sin x \sin h - x \cos x - h \cos x}{h x \left( x + h \right)}\]
\[ = \lim_{h \to 0} \frac{x \cos x \cos h - x \cos x - x \sin x \sin h - h \cos x}{h x \left( x + h \right)}\]
\[ = x\cos x \lim_{h \to 0} \frac{\cos h - 1}{h} - \frac{x\sin x}{x} \lim_{h \to 0} \frac{\sin h}{h} \lim_{h \to 0} \frac{1}{x + h} - \frac{\cos x}{x} \lim_{h \to 0} \frac{1}{x + h}\]
\[ = x \cos x \lim_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{\frac{h^2}{4}} \times \frac{h}{4} - \frac{x\sin x}{x} \lim_{h \to 0} \frac{\sin h}{h} \lim_{h \to 0} \frac{1}{x + h} - \frac{\cos x}{x} \lim_{h \to 0} \frac{1}{x + h} \left[ \because \lim_{h \to 0} \frac{\sin^2 \frac{h}{2}}{\frac{h^2}{4}} = \lim_{h \to 0} \frac{\sin \frac{h}{2}}{\frac{h}{2}} \times \lim_{h \to 0} \frac{\sin \frac{h}{2}}{\frac{h}{2}} = 1 \times 1, i . e . 1 \right]\]
\[ = - x \cos x \lim_{h \to 0} \frac{h}{2} - \frac{x\sin x}{x} \lim_{h \to 0} \frac{\sin h}{h} \lim_{h \to 0} \frac{1}{x + h} - \frac{\cos x}{x} \lim_{h \to 0} \frac{1}{x + h}\]
\[ = - x \cos x \times 0 - \sin x \left( 1 \right)\frac{1}{x} - \frac{\cos x}{x}\frac{1}{x}\]
\[ = 0 - \frac{\sin x}{x} - \frac{\cos x}{x^2}\]
\[ = - \frac{\sin x}{x} - \frac{\cos x}{x^2}\]
\[ = \frac{- x \sin x - \cos x}{x^2}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of 99x at x = 100.
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)n
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 + cos x)/(sin x - cos x)`
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):
`(a + bsin x)/(c + dcosx)`
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
k xn
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
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 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:
\[3^{x^2}\]
tan 2x
\[\sqrt{\tan x}\]
\[\sin \sqrt{2x}\]
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
x2 ex log x
sin x cos x
\[\frac{x^2 \cos\frac{\pi}{4}}{\sin x}\]
x−3 (5 + 3x)
(ax + b)n (cx + d)n
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
Mark the correct alternative in of the following:
If \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
Mark the correct alternative in of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
