Advertisements
Advertisements
Question
Find the derivative of cos x from first principle.
Solution
Let f(x) = cos x
∴ f(x + h) = cos (x + h)
f(x + h) − f(x) = cos (x + h) − cos x
= `lim_(h->0)[(-2 sin ((x + h + x)/2) sin ((x + h - x)/2))/h]`
= `lim_(h->0) [(-2sin((2x + h)/2) sin(h/2))/h]`
= `lim_(h->0)[(-2 sin (x + h/2) sin (h/2))/h]`
∴ f'(x) = `lim_(h → 0) (f(x + h) - f(x))/h`
= `lim_(h → 0)(-2sin (x + h/2) sin h/2)/h`
= `lim_(h → 0)[-sin(x+ h/2)]((sin h/2)/(h/2))`
= `- sinx` ...........`[∵ lim_(h → 0) (sin h/2)/(h/2) = 1]`
Hence, `d/dx cos x = - sin x`
APPEARS IN
RELATED QUESTIONS
For some constants a and b, find the derivative of (x – a) (x – b).
Find the derivative of `(x^n - a^n)/(x -a)` for some constant a.
Find the derivative of the following function.
sin x cos x
Find the derivative of the following function:
5 sec x + 4 cos 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):
(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 + cos x)(x - tan 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):
`(4x + 5sin x)/(3x + 7cos 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/(1 + tan 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 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/(sin^n 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) = `1/x` by first principle.
Find the derivative of `cosx/(1 + sinx)`
`(x + 1/x)^3`
`(3x + 4)/(5x^2 - 7x + 9)`
`(x^5 - cosx)/sinx`
(sin x + cos x)2
(2x – 7)2 (3x + 5)3
x2 sin x + cos 2x
sin3x cos3x
If `y = sqrt(x) + 1/sqrt(x)`, then`(dy)/(dx)` at x = 1 is ______.
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 = 1 + x/(1!) + x^2/(2!) + x^3/(3!) + ...,` then `(dy)/(dx)` = ______.
