Question

How do you find the derivative of `(cos x)` using the limit definition?

Answer 1

See the explanation section below.

Explanation:

We'll need the following facts:

From trigonometry:
`cos(A+B) = cosAcosB-sinAsinB`

Fundamental trigonometric limits:

`lim_(theta rarr0) sin theta /theta = 1`

`lim_(theta rarr0) (cos theta - 1) /theta = 0`

And here we go:

`f(x) = cosx`

`f'(x) = lim_(hrarr0)(cos(x+h)-cosx)/h`

`= lim_(hrarr0)(cosxcosh-sinxsinh-cosx)/h`

`= lim_(hrarr0)(cosxcosh-cosx-sinxsinh)/h`

`= lim_(hrarr0)((cosxcosh-cosx)/h-(sinxsinh)/h)`

`= lim_(hrarr0)(cosx(cosh-1)/h-sinx(sinh)/h)`

`= cosx(lim_(hrarr0)(cosh-1)/h)-sinx(lim_(hrarr0)(sinh)/h)`

`= cosx(0)-sinx(1)`

`= -sinx`