f #note
chain rule
Another common way to combine functions together is composition, that is using the output of one function as the input to another.
For example, if we compose inside the function , we get the function .
Now say we want to find the derivative of this function.
First lets think of the tiny nudge in the value of , .
The derivate of the function is normaly , but in this case the tiny nudge to is not , its , and so our derivative is . We can then take the derivative of and finaly obtain :
we can divide by to obtain , so :
We can notice a patern here, for a function composed with another, we first compose the derivative of the outer function with the inner function, and then multiply by the derivative of the inner function, like this :