Question

How do you find the derivative of `cos(pi*x^2)`?

Answer 1

The derivative is `- 2 sin(pi x^2) * pi x`.

Explanation:

To build the derivative of `cos(pi * x^2)`, you need to use the chain rule.

If `f(u) = cos(u)` and `u(x) = pi x^2`, the derivative is:

`f'(x) = f'(u(x)) * u'(x)`

So, you need to compute the derivatives of `f(u)` and `u(x)`:

`f(u) = cos(u) color(white)(xxxx) => f'(u) = - sin (u)`
`u(x) = pi x^2 color(white)(xxxxxx) => u'(x) = 2 pi x`

Now we can compute the derivative:

`f'(x) = - sin( pi x^2) * 2 pi x = - 2 sin(pi x^2) * pi x`