Advertisements
Advertisements
Question
\[\frac{\sec x - 1}{\sec x + 1}\]
Advertisements
Solution
\[\text{ Then }, u' = \sec x tan x; v' = \sec x \tan x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{sec x - 1}{sec x + 1} \right) = \frac{\left( \sec x + 1 \right)\sec x \tan x - \left( \sec x - 1 \right)\sec x \tan x}{\left( sec x + 1 \right)^2}\]
\[ = \frac{\sec^2 x \tan x + \sec x \tan x - \sec^2 x \tan x + \sec x \tan x}{\left( \sec x + 1 \right)^2}\]
\[ = \frac{2\sec x \tan x}{\left( \sec x + 1 \right)^2}\]
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):
(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):
(ax + b) (cx + d)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):
`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):
cosec x cot 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):
x4 (5 sin x – 3 cos x)
Find the derivative of the following function at the indicated point:
\[\frac{2}{x}\]
\[\frac{x^2 + 1}{x}\]
\[\frac{2x + 3}{x - 2}\]
Differentiate each of the following from first principle:
e−x
Differentiate of the following from first principle:
(−x)−1
Differentiate of the following from first principle:
x cos x
\[\sin \sqrt{2x}\]
\[\cos \sqrt{x}\]
\[\tan \sqrt{x}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
2 sec x + 3 cot x − 4 tan x
a0 xn + a1 xn−1 + a2 xn−2 + ... + an−1 x + an.
\[\frac{1}{\sin x} + 2^{x + 3} + \frac{4}{\log_x 3}\]
\[\text{ If } y = \left( \sin\frac{x}{2} + \cos\frac{x}{2} \right)^2 , \text{ find } \frac{dy}{dx} at x = \frac{\pi}{6} .\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
xn tan x
(x3 + x2 + 1) sin x
(x sin x + cos x) (x cos x − sin x)
x3 ex cos 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.
(3 sec x − 4 cosec x) (−2 sin x + 5 cos x)
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{x \tan x}{\sec x + \tan x}\]
\[\frac{2^x \cot x}{\sqrt{x}}\]
\[\frac{3^x}{x + \tan x}\]
If x < 2, then write the value of \[\frac{d}{dx}(\sqrt{x^2 - 4x + 4)}\]
Write the value of \[\frac{d}{dx}\left( x \left| x \right| \right)\]
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 \[y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}}\] then \[\frac{dy}{dx} =\]
Mark the correct alternative in each of the following:
If\[f\left( x \right) = \frac{x^n - a^n}{x - a}\] then \[f'\left( a \right)\]
