Using Chain Rule to Derive a Function of a Function

Question

So I have an expression z is equal to x of y and it is denoted as:

$z = x\circ y$

The properties of x and y are the function values and their derivative values;

$x(0) = 1$

$y(0) = 2$

$x '(0) = 3$

$x '(2) = 4$

$y '(0) = 5$

$y '(2) = 6$

I am to find $z '(0)$. How must I go about doing this? It is a different concept from what I have been learning.

Answer 1

Since you've mentioned chain rule, you should recall that

$$z'(t)=y'(t)x'(y(t))\implies z'(0)=y'(0)x'(y(0))=5x'(2)=5\times4$$

Answer 2

$$z'(t) = ((x' \circ y) \cdot y')(t) = x'(y(t))y'(t)$$ So $z'(0) = ?$