Advertisements
Advertisements
Question
\[\sqrt{\tan x}\]
Advertisements
Solution
\[ \frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}\]
\[ = \lim_{h \to 0} \frac{\sqrt{\tan\left( x + h \right)} - \sqrt{\tan x}}{h} \times \frac{\sqrt{\tan\left( x + h \right)} + \sqrt{\tan x}}{\sqrt{\tan\left( x + h \right)} + \sqrt{\tan x}}\]
\[ = \lim_{h \to 0} \frac{\tan\left( x + h \right) - \tan x}{h\left( \sqrt{\tan\left( x + h \right)} + \sqrt{\tan x} \right)}\]
\[ = \lim_{h \to 0} \frac{\frac{\sin \left( x + h \right)}{\cos \left( x + h \right)} - \frac{\sin x}{\cos x}}{h\left( \sqrt{\tan\left( x + h \right)} + \sqrt{\tan x} \right)}\]
\[ = \lim_{h \to 0} \frac{\sin \left( x + h \right) \cos x - \cos(x + h) \sin x}{h\left( \sqrt{\tan\left( x + h \right)} + \sqrt{\tan x} \right) \cos \left( x + h \right) \cos x}\]
\[ = \lim_{h \to 0} \frac{\sin h}{h\left( \sqrt{\tan\left( x + h \right)} + \sqrt{\tan x} \right) \cos \left( x + h \right) \cos x} \]
\[ = \lim_{h \to 0} \frac{\sin h}{h} \lim_{h \to 0} \frac{1}{\left( \sqrt{\tan\left( x + h \right)} + \sqrt{\tan x} \right) \cos \left( x + h \right) \cos x}\]
\[ = \left( 1 \right)\frac{1}{2 \sqrt{\tan x} \cos^2 x}\]
\[ = \frac{\sec^2 x}{2 \sqrt{\tan x}}\]
APPEARS IN
RELATED QUESTIONS
Find the derivative of x2 – 2 at x = 10.
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/(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):
`4sqrtx - 2`
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 f (x) = tan x at x = 0
k xn
\[\frac{1}{\sqrt{3 - x}}\]
\[\sqrt{2 x^2 + 1}\]
Differentiate each of the following from first principle:
\[\frac{\sin x}{x}\]
Differentiate each of the following from first principle:
x2 sin x
Differentiate each of the following from first principle:
sin x + cos x
Differentiate each of the following from first principle:
x2 ex
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
Differentiate each of the following from first principle:
\[a^\sqrt{x}\]
tan 2x
\[\sin \sqrt{2x}\]
\[\frac{x^3}{3} - 2\sqrt{x} + \frac{5}{x^2}\]
\[\frac{( x^3 + 1)(x - 2)}{x^2}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
\[\text{ If } y = \frac{2 x^9}{3} - \frac{5}{7} x^7 + 6 x^3 - x, \text{ find } \frac{dy}{dx} at x = 1 .\]
For the function \[f(x) = \frac{x^{100}}{100} + \frac{x^{99}}{99} + . . . + \frac{x^2}{2} + x + 1 .\]
x2 sin x log x
x−3 (5 + 3x)
(ax + b) (a + d)2
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{x \sin x}{1 + \cos x}\]
\[\frac{a + \sin x}{1 + a \sin x}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{x^5 - \cos x}{\sin x}\]
\[\frac{x + \cos x}{\tan x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Mark the correct alternative in of the following:
If \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Find the derivative of f(x) = tan(ax + b), by first principle.
