00:02
Let's find the derivative of f of x, and f of x is a product.
00:07
The first factor is 2x minus 3 to the 4th, and the second factor is x squared plus x plus 1 to the 5th.
00:14
So we're going to use the product rule.
00:17
So we start with the first, 2x minus 3 to the 4th, times the derivative of the second.
00:23
And the second is a composite, so we're going to need to use the chain rule.
00:27
So the derivative of the outside, the outside would be the fifth power function.
00:31
So we would bring down the 5 and raise the inside to the fourth.
00:36
So that takes care of the derivative of the outside.
00:39
Now we multiply it by the derivative of the inside.
00:42
The derivative of x squared plus x plus 1 would be 2x plus 1.
00:46
So what we've done so far is the first times the derivative of the second.
00:51
Now we're going to do plus the second x squared plus x plus 1 to the 5th times the derivative of the first.
00:59
And the first is also a composite.
01:02
So we're going to use the chain rule to find the derivative of 2x minus 3 to the 4th.
01:07
The outside function is the 4th power function.
01:10
So we bring down the 4 and we raise 2x minus 3 to the 3.
01:14
And now we multiply by the derivative of the inside.
01:17
The derivative of 2x minus 3 is 2.
01:20
Okay, so we have our derivative and now it's a matter of simplifying.
01:24
So this entire first part is our first term.
01:28
And this entire second part is our second term.
01:32
Squeeze my one back in there.
01:34
Let's see if we have any common factors that we can factor out of both terms.
01:39
It looks like both terms have 2x minus 3.
01:42
This one has it to the 4th.
01:44
This one has it to the 3rd.
01:46
So we can factor out 2x minus 3 to the 3 from both of them.
01:52
And it looks like both terms have x squared plus x plus 1.
01:56
This one has it to the 4th.
01:59
This one has it to the fifth...