Question

Find the derivative of the following function.

sin x cos x 

Answer 1

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