Advertisements
Advertisements
Question
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Advertisements
Solution
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sqrt{\sin \left( 2x + 2h \right)} - \sqrt{\sin 2x}}{h} \times \frac{\sqrt{\sin \left( 2x + 2h \right)} + \sqrt{\sin 2x}}{\sqrt{\sin \left( 2x + 2h \right)} + \sqrt{\sin 2x}}\]
\[ = \lim_{h \to 0} \frac{\sin \left( 2x + 2h \right) - \sin 2x}{h \left( \sqrt{\sin \left( 2x + 2h \right)} + \sqrt{\sin 2x} \right)}\]
\[\text{ We have }:\]
\[sin C-sin D= 2 cos\left( \frac{C + D}{2} \right)\sin\left( \frac{C - D}{2} \right)\]
\[ = \lim_{h \to 0} \frac{2 \cos \left( \frac{2x + 2h + 2x}{2} \right) \sin \left( \frac{2x + 2h - 2x}{2} \right)}{h \left( \sqrt{\sin \left( 2x + 2h \right)} + \sqrt{\sin 2x} \right)}\]
\[ = \lim_{h \to 0} \frac{2 \cos \left( 2x + h \right) \sin h}{h \left( \sqrt{\sin \left( 2x + 2h \right)} + \sqrt{\sin 2x} \right)}\]
\[ = \lim_{h \to 0} 2 \cos \left( 2x + h \right) \lim_{h \to 0} \frac{\sin h}{h} \lim_{h \to 0} \frac{1}{\left( \sqrt{\sin \left( 2x + 2h \right)} + \sqrt{\sin 2x} \right)} \]
\[ = 2 \cos 2x \left( 1 \right) \frac{1}{\sqrt{\sin 2x} + \sqrt{\sin 2x}}\]
\[ = \frac{2 \cos 2x}{2\sqrt{\sin 2x}}\]
\[ = \frac{\cos 2x}{\sqrt{\sin 2x}}\]
\[\]
APPEARS IN
RELATED QUESTIONS
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):
`(ax + b)/(cx + d)`
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):
`(ax + b)/(px^2 + qx + r)`
Find the derivative of the following function at the indicated point:
\[\frac{x^2 - 1}{x}\]
\[\frac{x + 2}{3x + 5}\]
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
eax + b
x ex
Differentiate of the following from first principle:
sin (x + 1)
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
\[e^\sqrt{ax + b}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
(2x2 + 1) (3x + 2)
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
xn tan x
xn loga x
(x3 + x2 + 1) sin x
(1 − 2 tan x) (5 + 4 sin x)
x3 ex cos 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)n (cx + d)n
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x^2 - x + 1}{x^2 + x + 1}\]
\[\frac{1 + 3^x}{1 - 3^x}\]
\[\frac{x}{\sin^n x}\]
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.
If f (x) = \[\log_{x_2}\]write the value of f' (x).
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
Find the derivative of x2 cosx.
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
