Advertisements
Advertisements
प्रश्न
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):
`(sec x - 1)/(sec x + 1)`
Advertisements
उत्तर
Let f(x) = `(sec x - 1)/(sec x + 1)`
f(x) = `(1/cos x -1)/(1/cos x +1)`
= `(1 - cos x)/(1 + cos x)`
By quotient rule,
f'(x) = `((1 + cosx)d/dx(1 - cosx)-(1 - cos x)d/dx(1 + cos x))/((1 + cos x^2))`
= `((1 + cos x) (sin x) - (1 - cos x) (-sin x))/((1 + cos x)^2)`
= `(sin x + cos x sin x + sin x - sin x cos x) /(1 + cos x)^2`
= `(2 sin x)/(1 + cos x)^2`
= `(2 sin x)/(1 + 1/sec x)^2 = (2 sin x)/((sec x + 1)^2/(sec^2 x))`
= `(2 sin x sec^2x)/ (secx+1)^2`
= `((2 sin x)/(cos x)sec x)/(sec x + 1)^2`
= `(2sec x tan x)/(sec x + 1)^2`
APPEARS IN
संबंधित प्रश्न
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):
`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):
(ax + b)n
Find the derivative of f (x) = 3x at x = 2
Find the derivative of f (x) x at x = 1
(x2 + 1) (x − 5)
Differentiate of the following from first principle:
e3x
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:
\[e^\sqrt{2x}\]
\[\sin \sqrt{2x}\]
\[\tan \sqrt{x}\]
\[\frac{a \cos x + b \sin x + c}{\sin x}\]
\[\frac{(x + 5)(2 x^2 - 1)}{x}\]
\[If y = \sqrt{\frac{x}{a}} + \sqrt{\frac{a}{x}}, \text{ prove that } 2xy\frac{dy}{dx} = \left( \frac{x}{a} - \frac{a}{x} \right)\]
sin x cos x
\[\frac{2^x \cot x}{\sqrt{x}}\]
(x sin x + cos x ) (ex + x2 log x)
(2x2 − 3) sin 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.
(3x2 + 2)2
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.
(x + 2) (x + 3)
(ax + b)n (cx + d)n
\[\frac{e^x - \tan x}{\cot x - x^n}\]
\[\frac{x}{1 + \tan x}\]
\[\frac{e^x + \sin x}{1 + \log x}\]
\[\frac{\sqrt{a} + \sqrt{x}}{\sqrt{a} - \sqrt{x}}\]
\[\frac{x^5 - \cos x}{\sin x}\]
Write the value of \[\frac{d}{dx} \left\{ \left( x + \left| x \right| \right) \left| x \right| \right\}\]
If f (x) = |x| + |x−1|, write the value of \[\frac{d}{dx}\left( f (x) \right)\]
Write the value of \[\frac{d}{dx}\left( \log \left| x \right| \right)\]
Mark the correct alternative in of the following:
Let f(x) = x − [x], x ∈ R, then \[f'\left( \frac{1}{2} \right)\]
Mark the correct alternative in of the following:
If\[f\left( x \right) = 1 - x + x^2 - x^3 + . . . - x^{99} + x^{100}\]then \[f'\left( 1 \right)\]
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 of the following:
If \[f\left( x \right) = x^{100} + x^{99} + . . . + x + 1\] then \[f'\left( 1 \right)\] is equal to
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
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
Mark the correct alternative in of the following:
If f(x) = x sinx, then \[f'\left( \frac{\pi}{2} \right) =\]
Find the derivative of f(x) = tan(ax + b), by first principle.
`(a + b sin x)/(c + d cos x)`
