Question

How do you find the derivative of `f(x) = cos(pi/2)x` using the chain rule?

Answer 1

For the function formulated in the question, the derivative is

`f'(x) = 0`.

However, the question might have meant this derivative instead:

`f'(x) = - pi/2 sin(pi/2 x)`

Explanation:

If you meant the function the way that you had typed it, then `cos (pi/2)` is just a coefficient of `x` and

`cos(pi/2) = 0`.

Thus,

`f(x) = cos(pi/2) * x = 0 * x = 0`

You don't need the chain rule for this one since the derivative of `f(x) = 0` is `f'(x) = 0`.

======================

You might have also meant

`f(x) = cos(pi/2 x)`

instead.

In this case, you can apply the chain rule as follows:

`f(x) = cos(color(blue)(u)) " where " color(blue)(u) = pi/2 x`

The derivative of `f(x)` is the derivative of `cos u` multiplied with the derivative of `u`:

`f'(x) = [cos u]' * u'`

It holds

`[cos u]' = - sin u = - sin (pi/2 x)`

`[u]' = [pi/2 x ]' = pi/2`

Thus, your derivative is

`f'(x) = [cos u]' * u' = - pi/2 sin(pi/2 x)`