Advertisements
Advertisements
प्रश्न
\[\sin \sqrt{2x}\]
Advertisements
उत्तर
\[\text{ Let } f(x) = \sin\sqrt{2x} \]
\[\text{ Thus, we have }: \]
\[ f(x + h) = \sin\sqrt{2\left( x + h \right)}\]
\[\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sin \sqrt{2x + 2h} - \sin \sqrt{2x}}{h}\]
\[\text{ We know }:\]
\[sin C- sin D=2 sin\left( \frac{C - D}{2} \right)\cos\left( \frac{C + D}{2} \right)\]
\[ = \lim_{h \to 0} \frac{2 \sin\left( \sqrt{2x + 2h} - \sqrt{2x} \right) \cos\left( \sqrt{2x + 2h} - \sqrt{2x} \right)}{h}\]
\[ = \lim_{h \to 0} \frac{2 \times 2 \sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right) \cos\left( \frac{\sqrt{2x + 2h} + \sqrt{2x}}{2} \right)}{2h + 2x - 2x}\]
\[ = \lim_{h \to 0} \frac{2 \times 2 \sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right) \cos\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{\left( \sqrt{2x + 2h} - \sqrt{2x} \right)\sqrt{2x + 2h} + \sqrt{2x}}\]
\[ = \lim_{h \to 0} \frac{2 \times 2 \sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right) \cos\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{2 \times \left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)\left( \sqrt{2x + 2h} + \sqrt{2x} \right)}\]
\[ = \lim_{h \to 0} \frac{\sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)} \lim_{h \to 0} \frac{2\cos \left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{\sqrt{2x + 2h} + \sqrt{2x}} \]
\[ = 1 \times \frac{2\cos\sqrt{2x}}{2\sqrt{2x}} \left[ \because \lim_{h \to 0} \frac{\sin\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)}{\left( \frac{\sqrt{2x + 2h} - \sqrt{2x}}{2} \right)} = 1 \right]\]
\[ = \frac{\cos\sqrt{2x}}{\sqrt{2x}}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of x2 – 2 at x = 10.
Find the derivative of `2x - 3/4`
Find the derivative of `2/(x + 1) - x^2/(3x -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):
`1/(ax^2 + bx + c)`
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`
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):
sin (x + a)
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):
`(sin x + cos x)/(sin x - cos 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
\[\sqrt{2 x^2 + 1}\]
\[\frac{2x + 3}{x - 2}\]
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
x sin x
Differentiate of the following from first principle:
sin (2x − 3)
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
\[\sqrt{\sin (3x + 1)}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
x4 − 2 sin x + 3 cos x
\[\left( x + \frac{1}{x} \right)\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
cos (x + a)
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
x3 sin x
x3 ex
xn tan x
xn loga x
(1 − 2 tan x) (5 + 4 sin x)
(2x2 − 3) sin x
\[\frac{2x - 1}{x^2 + 1}\]
\[\frac{x + e^x}{1 + \log x}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{1}{a x^2 + bx + c}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
If f (1) = 1, f' (1) = 2, then write the value of \[\lim_{x \to 1} \frac{\sqrt{f (x)} - 1}{\sqrt{x} - 1}\]
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
Find the derivative of f(x) = tan(ax + b), by first principle.
