Advertisements
Advertisements
Question
Find the derivative of the following function.
sin x cos x
Solution
Let f (x) = sin x cos x. Accordingly, from the first principle,
f'(x) = `lim_(h->0) (f(x + h) - f(x))/h`
= `lim_(h->0) (sin (x + h) cos(x + h) - sin x cos x)/h`
= `lim_(h->0)1/(2h)[2sin(x + h) cos(x + h) -2 sin x cos x]`
= `lim_(h->0)1/(2h)[sin2 (x + h)-sin2x]`
= `lim_(h-_0)1/(2h)[2cos (2x + 2h + 2x)/2 . sin (2x + 2h -2x)/2]`
= `lim_(h->0)1/h [cos (4x + 2h)/2 sin (2h)/h]`
= `lim_(h->0)1/h [cos (2x + h) sin h]`
= `lim_(h->0) cos (2x + h). lim_(h->0)sin h/h]`
= cos (2x + 0).1
= cos 2x
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.
For some constants a and b, find the derivative of `(x - a)/(x - b)`.
Find the derivative of cos x from first principle.
Find the derivative of the following function:
5 sec x + 4 cos x
Find the derivative of the following function:
cosec 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):
`(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/(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/(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) = 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.
`(x^4 + x^3 + x^2 + 1)/x`
`(x + 1/x)^3`
`(x^2 cos pi/4)/sinx`
(sin x + cos x)2
(2x – 7)2 (3x + 5)3
x2 sin x + cos 2x
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 = (sin(x + 9))/cosx` then `(dy)/(dx)` at x = 0 is ______.
If `f(x) = 1 + x + x^2/2 + ... + x^100/100`, then f'(1) is equal to ______.
If `f(x) = x^100 + x^99 .... + x + 1`, then f'(1) is equal to ______.
If `y = 1 + x/(1!) + x^2/(2!) + x^3/(3!) + ...,` then `(dy)/(dx)` = ______.
