00:01
Okay, so we have this function f of x equals x to the 5 times e to the x, and we want to find the first two derivatives.
00:09
So what we're going to use here is the product raw.
00:11
So we have a product of two functions, x to the 5, e to the x, so we need to use the product raw.
00:17
So if we want to find the derivative f prime, so this is just d by dx of x to the 5 times e to the x.
00:25
By the product rule, what we should do is differentiate the first term, which is x to the 5 and leave the second term alone, and then add the first term left alone times the derivative of the second term.
00:39
So this is what we need to do.
00:41
This is the product raw.
00:42
So d by d x of x to the power 5 is 5x to the power of 4.
00:48
And then we also have this e to the x term, plus x to the 5.
00:54
And the derivative of e to the x is just e to the x.
00:58
The exponential is its own derivative.
01:01
So this is just e to the x.
01:03
So this is our answer.
01:04
We could simplify this slightly if we wanted by bringing out the factor of e to the x, but it's not essential...