Advertisements
Advertisements
प्रश्न
(x sin x + cos x) (x cos x − sin x)
Advertisements
उत्तर
\[u = x \sin x + \cos x; v = x \cos x - \sin x\]
\[u' = x \cos x + \sin x - \sin x = x \cos x ; v' = - x \sin x + \cos x - \cos x = - x \sin x\]
\[ \]
\[\text{ Using the product rule }:\]
\[\frac{d}{dx}\left( uv \right) = uv' + vu'\]
\[\frac{d}{dx}\left[ \left( x \sin x + \cos x \right)\left( x \cos x - \sin x \right) \right] = \left( x \sin x + \cos x \right)\left( - x \sin x \right) + \left( x \cos x - \sin x \right)\left( x \cos x \right)\]
\[ = - x^2 \sin^2 x - x \cos x \sin x + x^2 \cos^2 x - x \cos x \sin x \]
\[ = x^2 \left( \cos^2 x - \sin^2 x \right) - x\left( 2 \sin x \cos x \right)\]
\[ = x^2 \cos \left( 2x \right) - x\left( \sin \left( 2x \right) \right)\]
\[ = x \left[ x \cos \left( 2x \right) - \sin \left( 2x \right) \right]\]
APPEARS IN
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 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):
`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):
x4 (5 sin x – 3 cos x)
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:
2 cos x at x =\[\frac{\pi}{2}\]
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{1}{\sqrt{3 - x}}\]
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
x ex
Differentiate of the following from first principle:
(−x)−1
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:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
\[\tan \sqrt{x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
If for f (x) = λ x2 + μ x + 12, f' (4) = 15 and f' (2) = 11, then find λ and μ.
x3 ex
\[\frac{2^x \cot x}{\sqrt{x}}\]
(1 +x2) cos 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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
(ax + b) (a + d)2
(ax + b)n (cx + d)n
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{\sin x - x \cos x}{x \sin x + \cos x}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
If f (x) = \[\log_{x_2}\]write the value of f' (x).
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
Find the derivative of f(x) = tan(ax + b), by first principle.
