Advertisements
Advertisements
Question
\[\sin \sqrt{2x}\]
Solution
\[\text{ Let } f(x) = \sin\sqrt{2x} \]
\[\text{ Thus, we have }: \]
\[ f(x + h) = \sin\sqrt{2\left( x + h \right)}\]
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin \sqrt{2x + 2h} - \sin \sqrt{2x}}{h}\]
\[\text{ We know }:\]
\[sin C- sin D=2 sin\left( \frac{C - D}{2} \right)\cos\left( \frac{C + D}{2} \right)\]
\[ = \lim_{h \to 0} \frac{2 \sin\left( \sqrt{2x + 2h} - \sqrt{2x} \right) \cos\left( \sqrt{2x + 2h} - \sqrt{2x} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{2 \times 2 \sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right) \cos\left( \frac{\sqrt{2x + 2h} + \sqrt{2x}}{2} \right)}{2h + 2x - 2x}\]
\[ = \lim_{h \to 0} \frac{2 \times 2 \sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right) \cos\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{\left( \sqrt{2x + 2h} - \sqrt{2x} \right)\sqrt{2x + 2h} + \sqrt{2x}}\]
\[ = \lim_{h \to 0} \frac{2 \times 2 \sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right) \cos\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{2 \times \left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)\left( \sqrt{2x + 2h} + \sqrt{2x} \right)}\]
\[ = \lim_{h \to 0} \frac{\sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)} \lim_{h \to 0} \frac{2\cos \left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{\sqrt{2x + 2h} + \sqrt{2x}} \]
\[ = 1 \times \frac{2\cos\sqrt{2x}}{2\sqrt{2x}} \left[ \because \lim_{h \to 0} \frac{\sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)} = 1 \right]\]
\[ = \frac{\cos\sqrt{2x}}{\sqrt{2x}}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x–3 (5 + 3x).
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):
`a/x^4 = b/x^2 + cos x`
\[\frac{2}{x}\]
x ex
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
x2 sin x
\[\sqrt{\tan x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
(2x2 + 1) (3x + 2)
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
cos (x + a)
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
xn loga x
(x sin x + cos x) (x cos x − sin x)
logx2 x
\[e^x \log \sqrt{x} \tan x\]
x4 (5 sin x − 3 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.
(3x2 + 2)2
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{a + b \sin x}{c + d \cos x}\]
Write the value of \[\lim_{x \to a} \frac{x f (a) - a f (x)}{x - a}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
Write the value of the derivative of f (x) = |x − 1| + |x − 3| at x = 2.
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 \[f\left( x \right) = \frac{x - 4}{2\sqrt{x}}\]
Mark the correct alternative in of the following:
If \[y = \frac{\sin\left( x + 9 \right)}{\cos x}\] then \[\frac{dy}{dx}\] at x = 0 is
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
