Question

Find the derivative of cos x from first principle.

Answer 1

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`