Question

Find the derivative of the following function:

5sin x – 6cos x + 7

Answer 1

Let f (x) = 5sin x – 6cos x + 7. Accordingly, from the first principle,

`f'(x)` = `lim_(h->0)(f(x + h) - f(x))/h`

= `lim_(h->0)1/h [5 sin (x + h) - 6cos (x + h) + 7 - 5sinx + 6 cos x - 7]`

= `lim_(h->0)1/h [5{sin(x + h)- sinx} -6{cos (x + h)-cosx}]`

= `5lim_(h->0)1/h[sin(x + h)-sinx] -6 lim_(h->0)1/h[cos (x + h) - cos x]`

= `5lim_(h->0)1/h [2 cos  ((x + h + x)/2) sin  ((x + h - x)/2)] - 6lim_(h->0) (cos x cos h - sin x sin h - cos x)/h`

= `5lim_(h->0)1/h [2 cos  ((2x + h)/h) sin  h/2] -6lim_(h->0)[(-cos x (1 - cos h) - sin x sin h)/h]`

= `5lim_(h->0) (cos  ((2x + h)/2) (sin  h/2)/(h/2)) - 6lim_(h->0) [(-cosx (1 - cos h))/h - (sin x sin h)/h]`

= `5 [lim_(h->0)((2x +h)/2)][lim_(h->0) (sin  h/2)/(h/2)] -6 [(- cos x) (lim_(h->0)(1 - cos h)/h) - sin x lim_(h->0) ((sin h)/h)]`

= 5 cosx. 1 - 6 [(-cos x) (0) -sinx.1]

= 5 cosx + 6 sin x