SECTION 3.3: The Chain Rule

If y = f(z) and z = g(t), then the derivative of y = f(g(t)) is given as
dy/dt = dy/dz . dz/dt as long as dz and dt do not equal 0.

In other words, the derivative of a composite function s the derivative of the outside function times the derivative of the inside function:
d/dt f (g (t)) = f ' (g (t)) . g ' (t)

The chain rule can also be found for larger formulae by delaing with the formula as a single variable. For example,

y = (2t+1)^4 can be differntiated first by assuming a variable z = (2t+1)
Therefore, y = z^4
y' = 4(z^3)*z' = 4((2t+1)^3)*2 = 8(2t+1)^3

Some standard differential identities are as follows:
d/dt (t^n) = n*(t^n-1)
d/dt (e^t) = e^t
d/dt (ln t) = 1/t
d/dt (e^kt) = k*(e^kt) where k is an integer constant

Similarly,
d/dt (z^n) = n*(z^n-1)*dz/dt
d/dt (e^z) = (e^z)*dz/dt
d/dt (ln z) = (1/z)*dz/dt

I found it difficult to understand the example of how to determine the derivative of a composite function from a graph. I couldn't understand how it was being calculated. The theory made sense but the visuals didn't.

Again, learning how to determine rates of changes of irregular slopes over smaller and smaller intervals will always be useful in analyzing future market trends in a subject like Economics.