Advertisements
Advertisements
Question
\[\cos \sqrt{x}\]
Advertisements
Solution
\[ \text{ Let } f(x) = \cos \sqrt{x} \]
\[\text{ Thus, we have }: \]
\[ f(x + h) = \cos \sqrt{x + h}\]
\[\frac{d}{dx}\left( f\left( x \right) \right) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\]
\[ = \lim_{h \to 0} \frac{\cos \sqrt{x + h} - \cos \sqrt{x}}{h}\]
\[\text{ We know }: \]
\[ \cos C - \cos D = - 2\sin\left( \frac{C + D}{2} \right) \sin\left( \frac{C - D}{2} \right)\]
\[ = \lim_{h \to 0} \frac{- 2\sin \left( \frac{\sqrt{x + h} + \sqrt{x}}{2} \right) \sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{- 2\sin \left( \frac{\sqrt{x + h} + \sqrt{x}}{2} \right) \sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{x + h - x}\]
\[ = \lim_{h \to 0} \frac{- 2\sin \left( \frac{\sqrt{x + h} + \sqrt{x}}{2} \right) \sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{2 \times \left( \sqrt{x + h} + \sqrt{x} \right)\frac{\left( \sqrt{x + h} - \sqrt{x} \right)}{2}}\]
\[ = \lim_{h \to 0} \frac{\sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{\frac{\sqrt{x + h} - \sqrt{x}}{2}} \lim_{h \to 0} \frac{- \sin\left( \frac{\sqrt{x + h} + \sqrt{x}}{2} \right)}{\sqrt{x + h} + \sqrt{x}} \]
\[ = 1 \times \frac{- \sin\sqrt{x}}{2\sqrt{x}} \left[ \because \lim_{h \to 0} \frac{\sin\left( \frac{\sqrt{x + h} - \sqrt{x}}{2} \right)}{\frac{\sqrt{x + h} - \sqrt{x}}{2}} = 1 \right]\]
\[ = \frac{- \sin\sqrt{x}}{2\sqrt{x}}\]
\[\]
APPEARS IN
RELATED QUESTIONS
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):
`(px^2 +qx + r)/(ax +b)`
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):
`(sec x - 1)/(sec 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):
sinn x
Find the derivative of the following function at the indicated point:
sin 2x at x =\[\frac{\pi}{2}\]
\[\frac{1}{x^3}\]
\[\frac{x^2 + 1}{x}\]
\[\frac{1}{\sqrt{3 - x}}\]
(x2 + 1) (x − 5)
(x2 + 1) (x − 5)
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
\[\cos\left( x - \frac{\pi}{8} \right)\]
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
\[\sqrt{\sin 2x}\]
Differentiate each of the following from first principle:
\[\frac{\cos x}{x}\]
3x + x3 + 33
ex log a + ea long x + ea log a
log3 x + 3 loge x + 2 tan x
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
2 sec x + 3 cot x − 4 tan x
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
cos (x + a)
\[\text{ If } y = \left( \frac{2 - 3 \cos x}{\sin x} \right), \text{ find } \frac{dy}{dx} at x = \frac{\pi}{4}\]
xn tan x
sin x cos x
(1 − 2 tan x) (5 + 4 sin x)
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{x}{\sin^n x}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (x) = \[\frac{x^2}{\left| x \right|},\text{ write }\frac{d}{dx}\left( f (x) \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
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
Mark the correct alternative in each of the following:
If\[y = \frac{\sin x + \cos x}{\sin x - \cos x}\] then \[\frac{dy}{dx}\]at x = 0 is
