Advertisements
Advertisements
Question
Find the derivative of the following function:
cosec x
Advertisements
Solution
Let f(x) = cosec x
and f(x + h) = cosec (x + h)
f(x + h) – f(x) = cosec (x + h) – cosec x
= `1/(sin (x + h)) - 1/(sin x)`
= `(sin x - sin (x + h))/(sin (x + h) sin x)`
= `(-2 cos(x + h/2) sin h/2)/(sin (x + h) sin x)`
`f'(x) = lim_(h → 0)(f(x + h) - f(x))/h`
= `lim_(h → 0)1/h. (-2cos (x + h/2) sin h/2)/(sin (x + h) sin x)`
= `lim_(h → 0)(- cos(x + h/2))/(sin (x + h)sin x). ((sin h/2)/(h/2))`
= `-cosx/(sin x. sin x)`
= `- 1/sin x. (cos x)/(sin x)`
= − cosec x cot x
APPEARS IN
RELATED QUESTIONS
For some constants a and b, find the derivative of (x – a) (x – b).
For some constants a and b, find the derivative of (ax2 + b)2.
Find the derivative of the following function.
sin x cos x
Find the derivative of the following function:
sec x
Find the derivative of the following function:
3cot x + 5cosec x
Find the derivative of the following function:
5sin x – 6cos x + 7
Find the derivative of the following function:
2tan x – 7sec 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):
(x2 + 1) 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):
(ax2 + sin x) (p + q 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):
`(x^2 cos (pi/4))/sin 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):
(x + sec x) (x – tan x)
Find the derivative of f(x) = ax + b, where a and b are non-zero constants, by first principle
Find the derivative of f(x) = ax2 + bx + c, where a, b and c are none-zero constant, by first principle.
Find the derivative of f(x) = x3, by first principle.
Find the derivative of f(x) = `1/x` by first principle.
Find the derivative of f(x) = sin x, by first principle.
Find the derivative of `cosx/(1 + sinx)`
`(x^4 + x^3 + x^2 + 1)/x`
`(3x + 4)/(5x^2 - 7x + 9)`
`(x^2 cos pi/4)/sinx`
(sin x + cos x)2
(2x – 7)2 (3x + 5)3
if `f(x) = (x - 4)/(2sqrt(x))`, then f'(1) is ______.
If `y = (1 + 1/x^2)/(1 - 1/x^2)` then `(dy)/(dx)` is ______.
If `y = (sin x + cos x)/(sin x - cos x)`, then `(dy)/(dx)` at x = 0 is ______.
If `f(x) = 1 + x + x^2/2 + ... + x^100/100`, then f'(1) is equal to ______.
