00:01
This problem wants us to use the chain rule to find the derivative of f of x equals 4 times e raised to the negative 3x to the 5th plus 7x to the 7th.
00:09
The derivative using the chain rule means that we need to take the derivative of almost what it seems like the outermost function that has the input that relates to another type of function that we could treat as its own separate function.
00:22
So this negative 3x to the 5th plus 7x to the 7th appears to be a function that is being the input of an outer function, where the outer function would be.
00:30
Just 4e to the x, where again, negative 3x of 5th plus 7x to the 7th is taking the place of x.
00:36
So to apply the chain rule, we're going to take the derivative of 4e to the input without changing anything about the input's derivative just yet.
00:45
And the derivative of e to the x is just the e to the x function over again.
00:49
So the derivative of 4e to negative 3x to the 5th plus 7x to the 7th before taking the derivative of the input would just be 4 times e raised to the negative 3x to the 5th.
01:00
Plus 7x to the 7th.
01:02
But then to apply the chain rule, we're going to multiply now by the derivative of the input of the function.
01:08
We just took the derivative up.
01:10
So we will take the derivative of negative 3x to the 5th plus 7x to the 7th by taking our 5, multiplying it by negative 3 to give us negative 15x to the 4th after we take one away from 5...