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 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):
sinn x
Find the derivative of f (x) x at x = 1
Find the derivative of the following function at the indicated point:
sin x at x =\[\frac{\pi}{2}\]
\[\frac{1}{\sqrt{x}}\]
\[\frac{1}{x^3}\]
\[\frac{x^2 + 1}{x}\]
(x + 2)3
Differentiate of the following from first principle:
x cos x
Differentiate each of the following from first principle:
\[e^{x^2 + 1}\]
Differentiate each of the following from first principle:
\[3^{x^2}\]
\[\left( \sqrt{x} + \frac{1}{\sqrt{x}} \right)^3\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
2 sec x + 3 cot x − 4 tan x
Find the slope of the tangent to the curve f (x) = 2x6 + x4 − 1 at x = 1.
x3 ex
x2 ex log x
xn tan x
x2 sin x log x
x5 (3 − 6x−9)
(ax + b) (a + d)2
\[\frac{a x^2 + bx + c}{p x^2 + qx + r}\]
\[\frac{e^x}{1 + x^2}\]
\[\frac{{10}^x}{\sin x}\]
\[\frac{4x + 5 \sin x}{3x + 7 \cos x}\]
\[\frac{p x^2 + qx + r}{ax + b}\]
\[\frac{ax + b}{p x^2 + qx + r}\]
\[\frac{1}{a x^2 + bx + c}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
If \[\frac{\pi}{2}\] then find \[\frac{d}{dx}\left( \sqrt{\frac{1 + \cos 2x}{2}} \right)\]
If f (x) = \[\log_{x_2}\]write the value of f' (x).
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
Find the derivative of 2x4 + x.
Find the derivative of x2 cosx.
Find the derivative of f(x) = tan(ax + b), by first principle.
Let f(x) = x – [x]; ∈ R, then f'`(1/2)` is ______.
