Advertisements
Advertisements
Question
Differentiate of the following from first principle:
x cos x
Advertisements
Solution
\[\left( x \right) \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h \right) \cos \left( x + h \right) - x \cos x}{h}\]
\[ = \lim_{h \to 0} \frac{\left( x + h \right)\left( \cos x \cos h - \sin x \sin h \right) - x \cos x}{h}\]
\[ = \lim_{h \to 0} \frac{x \cos x \cos h - x \sin x \sin h + h \cos x \cos h - h \sin x \sin h - x \cos x}{h}\]
\[ = \lim_{h \to 0} \frac{x \cos x \cos h - x \cos x - x \sin x \sin h + h \cos x \cos h - h \sin x \sin h}{h}\]
\[ = x \cos x \lim_{h \to 0} \frac{\left( \cos h - 1 \right)}{h} - x \sin x \lim_{h \to 0} \frac{\sin h}{h} + \cos x \lim_{h \to 0} \cos h + \sin x \lim_{h \to 0} \sin h\]
\[ = x \cos x \lim_{h \to 0} \frac{- 2 \sin^2 \frac{h}{2}}{\frac{h^2}{4}} \times \frac{h}{4} - x \sin x \left( 1 \right) + \cos x \left( 1 \right) + \sin x \left( 0 \right)\]
\[ = x\cos x \lim_{h \to 0} \frac{- h}{2} - x \sin x \left( 1 \right) + \cos x \left( 1 \right) + \sin x \left( 0 \right)\]
\[ = - x \cos x \left( 0 \right) - x \sin x + \cos x \]
\[ = - x \sin x + \cos x \]
\[ \]
\[\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of 99x at x = 100.
Find the derivative of x at x = 1.
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):
`(px+ q) (r/s + s)`
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 + 1/x)/(1- 1/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):
sinn x
Find the derivative of f (x) = x2 − 2 at x = 10
Find the derivative of the following function at the indicated point:
Find the derivative of the following function at the indicated point:
2 cos x at x =\[\frac{\pi}{2}\]
Differentiate of the following from first principle:
e3x
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\sin \sqrt{2x}\]
\[\tan \sqrt{x}\]
3x + x3 + 33
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
cos (x + a)
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
x3 ex
x5 ex + x6 log x
(x sin x + cos x) (x cos x − sin x)
(x sin x + cos x ) (ex + x2 log x)
(1 − 2 tan x) (5 + 4 sin x)
\[e^x \log \sqrt{x} \tan x\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{1}{a x^2 + bx + c}\]
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 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 \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
`(a + b sin x)/(c + d cos x)`
