Can you explain to me the calculus chain rule?
The chain rule is a fundamental rule in calculus that allows you to find the derivative of a composite function. A composite function is a function that is formed by applying one function to the output of another function. Mathematically, if you have two functions, say f(u) and g(x), then the composite function h(x) can be represented as:

h(x) = f(g(x))

The chain rule states that to find the derivative of the composite function h(x) with respect to x, you should take the derivative of the outer function f(u) with respect to its input u (which is g(x) in this case), and then multiply it by the derivative of the inner function g(x) with respect to x. Symbolically, it can be expressed as:

h'(x) = f'(g(x)) * g'(x)

Here, h'(x) is the derivative of h(x) with respect to x, f'(g(x)) is the derivative of f(u) with respect to u evaluated at u = g(x), and g'(x) is the derivative of g(x) with respect to x.

Let's go through a simple example to illustrate the chain rule:

Suppose we have two functions f(u) = u^2 and g(x) = 3x + 1, and we want to find the derivative of their composite function h(x) = f(g(x)) with respect to x.

First, we find the derivatives of f(u) and g(x) with respect to their respective variables:

f'(u) = 2u
g'(x) = 3

Now, we apply the chain rule:

h'(x) = f'(g(x)) * g'(x)

h'(x) = 2(g(x)) * 3

Since g(x) = 3x + 1:

h'(x) = 2(3x + 1) * 3

Finally, we have:

h'(x) = 6(3x + 1)

Let's further simplify and calculate the derivative h'(x) for the composite function h(x) = f(g(x)) where f(u) = u^2 and g(x) = 3x + 1. We previously found:

h'(x) = 6(3x + 1)

Now, we just need to distribute the 6:

h'(x) = 6 * 3x + 6 * 1
h'(x) = 18x + 6

So, the derivative of the composite function h(x) with respect to x is:

h'(x) = 18x + 6

This means that at any point x, the rate of change or the slope of the tangent line to the curve of h(x) is given by 18x + 6. The chain rule allows us to easily differentiate composite functions, which is particularly useful when dealing with more complex functions and applications in various fields such as physics, engineering, and economics.
Thanks. Can you please explain it again, but shorter?