Advertisements
Advertisements
Question
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
Advertisements
Solution
\[\text{ We have }: \]
\[f'\left( \frac{\pi}{2} \right) = \lim_{h \to 0} \frac{f\left( \frac{\pi}{2} + h \right) - f\left( \frac{\pi}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin2\left( \frac{\pi}{2} + h \right) - \sin2\left( \frac{\pi}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin(\pi + 2h) - 0}{h}\]
\[ = \lim_{h \to 0} \frac{- \sin2h}{h} \times \frac{2}{2} \]
\[ = \lim_{h \to 0} - \frac{\sin 2h}{2h} \times 2 \]
\[ = - 2\]
\[\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of `2x - 3/4`
Find the derivative of x–3 (5 + 3x).
Find the derivative of x5 (3 – 6x–9).
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
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):
`(a + bsin x)/(c + dcosx)`
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 f (x) = 3x at x = 2
Find the derivative of f (x) x at x = 1
Find the derivative of the following function at the indicated point:
\[\frac{2}{x}\]
\[\frac{1}{x^3}\]
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 1}{x + 2}\]
(x2 + 1) (x − 5)
(x2 + 1) (x − 5)
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
\[\sin \sqrt{2x}\]
\[\tan \sqrt{x}\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
x2 ex log x
xn tan x
(x3 + x2 + 1) sin x
\[\frac{2^x \cot x}{\sqrt{x}}\]
(1 − 2 tan x) (5 + 4 sin 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.
(x + 2) (x + 3)
\[\frac{x}{1 + \tan x}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{\sec x - 1}{\sec x + 1}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
If |x| < 1 and y = 1 + x + x2 + x3 + ..., then write the value of \[\frac{dy}{dx}\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
`(a + b sin x)/(c + d cos x)`
