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
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 x–3 (5 + 3x).
Find the derivative of x–4 (3 – 4x–5).
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):
(ax + b)n (cx + d)m
\[\frac{1}{x^3}\]
\[\frac{x + 2}{3x + 5}\]
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
eax + b
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
x4 − 2 sin x + 3 cos x
\[\frac{2 x^2 + 3x + 4}{x}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
xn tan x
(x3 + x2 + 1) sin x
x2 sin x log x
\[e^x \log \sqrt{x} \tan x\]
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
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.
(x + 2) (x + 3)
(ax + b) (a + d)2
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
\[\frac{1}{a x^2 + bx + c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
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 \[y = \sqrt{x} + \frac{1}{\sqrt{x}}\] then \[\frac{dy}{dx}\] at x = 1 is
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Find the derivative of x2 cosx.
`(a + b sin x)/(c + d cos x)`
