Advertisements
Advertisements
प्रश्न
\[\frac{x \tan x}{\sec x + \tan x}\]
Advertisements
उत्तर
\[\text{ Let } u = x \tan x; v = \sec x + \tan x\]
\[\text{ Then }, u' = x \sec^2 x + \tan x; v' = \sec x \tan x + \sec^2 x\]
\[\text{ Using the quotient rule }:\]
\[\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{vu' - uv'}{v^2}\]
\[\frac{d}{dx}\left( \frac{x\tan x}{\sec x + \tan x} \right) = \frac{\left( \sec x + \tan x \right)\left( x \sec^2 x + \tan x \right) - x \tan x\left( \sec x \tan x + \sec^2 x \right)}{\left( \sec x + \tan x \right)^2}\]
\[ = \frac{x \sec^3 x + x \sec^2 x\tan x + \sec x \tan x + \tan^2 x - x \sec x \tan^2 x - x \tan x \sec^2 x}{\left( \sec x + \tan x \right)^2}\]
\[ = \frac{\left( \sec x + \tan x \right)\left( x \sec^2 x + \tan x \right) - x \tan x \sec x\left( \sec x + \tan x \right)}{\left( \sec x + \tan x \right)^2}\]
\[ = \frac{x \sec^2 x + \tan x - x \tan x \sec x}{\sec x + \tan x}\]
\[ = \frac{x \sec x\left( \sec x - \tan x \right) + \tan x}{\sec x + \tan x}\]
APPEARS IN
संबंधित प्रश्न
Find the derivative of (5x3 + 3x – 1) (x – 1).
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):
`(1 + 1/x)/(1- 1/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):
`(ax + b)/(px^2 + qx + r)`
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):
x4 (5 sin x – 3 cos x)
Find the derivative of f (x) = 3x at x = 2
Find the derivative of the following function at the indicated point:
\[\frac{2}{x}\]
\[\frac{1}{\sqrt{x}}\]
\[\frac{1}{x^3}\]
k xn
\[\frac{1}{\sqrt{3 - x}}\]
\[\sqrt{2 x^2 + 1}\]
Differentiate of the following from first principle:
− x
Differentiate of the following from first principle:
(−x)−1
Differentiate each of the following from first principle:
\[e^\sqrt{2x}\]
tan 2x
\[\sqrt{\tan x}\]
x4 − 2 sin x + 3 cos x
log3 x + 3 loge x + 2 tan x
\[\log\left( \frac{1}{\sqrt{x}} \right) + 5 x^a - 3 a^x + \sqrt[3]{x^2} + 6 \sqrt[4]{x^{- 3}}\]
x3 sin x
xn loga x
x2 sin x log x
x5 ex + x6 log x
x−4 (3 − 4x−5)
\[\frac{x^2 + 1}{x + 1}\]
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{3^x}{x + \tan x}\]
\[\frac{x + \cos x}{\tan x}\]
Write the value of \[\lim_{x \to c} \frac{f(x) - f(c)}{x - c}\]
Write the derivative of f (x) = 3 |2 + x| at x = −3.
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 f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
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 ______.
