Advertisements
Advertisements
प्रश्न
Differentiate each of the following from first principle:
x2 sin x
Advertisements
उत्तर
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h \right)^2 \sin \left( x + h \right) - x^2 \sin x}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x^2 + h^2 + 2xh \right)\left( \sin x \cos h + \cos x \sin h \right) - x^2 \sin x}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 \sin x \cos h + x^2 \cos x \sin h + h^2 \sin x \cos h + h^2 \cos x \sin h + 2xh \sin x \cos h + 2xh \cos x \sin h - x^2 \sin x}{h}\]
\[ = \lim_{h \to 0} \frac{x^2 \sin x \cos h - x^2 \sin x + x^2 \cos x \sin h + h^2 \sin x \cos h + h^2 \cos x \sin h + 2xh \sin x \cos h + 2xh \cos x \sin h}{h}\]
\[ = x^2 \sin x \lim_{h \to 0} \frac{\cos h - 1}{h} + x^2 \cos x \lim_{h \to 0} \frac{\sin h}{h} + \sin x \lim_{h \to 0} h \cos h + \cos x \lim_{h \to 0} h \sin h + 2x \sin x \lim_{h \to 0} \cosh + 2x \cos x \lim_{h \to 0} \sin h\]
\[ = x^2 \sin x \lim_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{\frac{h^2}{4}} \times \frac{h}{4} + x^2 \cos x \lim_{h \to 0} \frac{\sin h}{h} + \sin x \lim_{h \to 0} h \cos h + \cos x \lim_{h \to 0} h \sin h + 2x \sin x \lim_{h \to 0} \cosh + 2x \cos x \lim_{h \to 0} \sin h \left[ \because \lim_{h \to 0} \frac{\sin^2 \frac{h}{2}}{\frac{h^2}{4}} = \lim_{h \to 0} \frac{\sin \frac{h}{2}}{\frac{h}{2}} \times \lim_{h \to 0} \frac{\sin \frac{h}{2}}{\frac{h}{2}} = 1 \times 1, i . e . 1 \right]\]
\[ = - x^2 \sin x \times \lim_{h \to 0} \frac{h}{2} + x^2 \cos x \lim_{h \to 0} \frac{\sin h}{h} + \sin x \lim_{h \to 0} h \cos h + \cos x \lim_{h \to 0} h \sin h + 2x \sin x \lim_{h \to 0} \cosh + 2x \cos x \lim_{h \to 0} \sin h \]
\[ = - x^2 \sin x \times 0 + x^2 \cos x \left( 1 \right) + \sin x \left( 0 \right) + \cos x \left( 0 \right) + 2x \sin x \left( 1 \right) + 2x \cos x \left( 0 \right)\]
\[ = 0 + x^2 \cos x + 2x \sin x\]
\[ = 0 + x^2 \cos x + 2x \sin x\]
\[ = x^2 \cos x + 2x \sin x\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x at x = 1.
Find the derivative of (5x3 + 3x – 1) (x – 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):
`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):
`4sqrtx - 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):
sin (x + a)
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) = 99x at x = 100
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{2}{x}\]
\[\frac{1}{\sqrt{x}}\]
x2 + x + 3
(x2 + 1) (x − 5)
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
x4 − 2 sin x + 3 cos x
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x5 ex + x6 log x
Differentiate in two ways, using product rule and otherwise, the function (1 + 2 tan x) (5 + 4 cos x). Verify that the answers are the same.
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
(ax + b)n (cx + d)n
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
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 \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
Mark the correct alternative in of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
Find the derivative of f(x) = tan(ax + b), by first principle.
