Evaluating function taken as input

Consider this toy example:

myDerivative[f_, x_] := D[f, {x, 2}]

This defines a function myDerivative that takes in a function f and a variable x and takes the second derivative of f, i.e. it returns $f”$. This works as expected:

myDerivative[x^2, x]   

2

However, consider a slightly different version:

myDerivative2[f_, x_] := D[f[x^2], {x, 2}]

The difference here is that before differentiation, the function f is composed with x^2. This particular composition here is irrelevant, it’s just to highlight that I have to somehow evaluate the function f with my function, I can’t just leave as f like before. This no longer works:

myDerivative2[x^2, x]

2[x^2] + 2 x Derivative[1][(2 x)][x^2] + 2 Derivative[1][(x^2)][x^2] + 2 x (Derivative[1][(2 x)][x^2] + 2 x ((x^2)^[Prime][Prime])[x^2])

It does, however, work if I supply a pure function, or just a head:

myDerivative2[#^2 &, x]
myDerivative2[Sin, x]

12 x^2
2 Cos[x^2] - 4 x^2 Sin[x^2]

How can I define myDerivative2 such that myDerivative[x^2, x] works as expected, without resorting to inputting a pure function? Google surprisingly didn’t provide an answer to this (maybe I don’t know the right terms), all that I could find used pure functions like #^2& or Sin (just the head).