Advertisements
Advertisements
Question
(x sin x + cos x) (x cos x − sin x)
Solution
\[u = x \sin x + \cos x; v = x \cos x - \sin x\]
\[u' = x \cos x + \sin x - \sin x = x \cos x ; v' = - x \sin x + \cos x - \cos x = - x \sin x\]
\[ \]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left[ \left( x \sin x + \cos x \right)\left( x \cos x - \sin x \right) \right] = \left( x \sin x + \cos x \right)\left( - x \sin x \right) + \left( x \cos x - \sin x \right)\left( x \cos x \right)\]
\[ = - x^2 \sin^2 x - x \cos x \sin x + x^2 \cos^2 x - x \cos x \sin x \]
\[ = x^2 \left( \cos^2 x - \sin^2 x \right) - x\left( 2 \sin x \cos x \right)\]
\[ = x^2 \cos \left( 2x \right) - x\left( \sin \left( 2x \right) \right)\]
\[ = x \left[ x \cos \left( 2x \right) - \sin \left( 2x \right) \right]\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of 99x at x = 100.
Find the derivative of x at x = 1.
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–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):
`cos x/(1 + sin 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):
`(sin(x + a))/ cos x`
Find the derivative of f (x) = tan x at x = 0
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
k xn
(x2 + 1) (x − 5)
\[\sqrt{2 x^2 + 1}\]
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
tan 2x
\[\tan \sqrt{x}\]
x4 − 2 sin x + 3 cos x
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x2 ex log x
(x3 + x2 + 1) sin x
x−3 (5 + 3x)
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.
(x + 2) (x + 3)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{1}{a x^2 + bx + c}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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
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
`(a + b sin x)/(c + d cos x)`
