00:01
We want to find the first and second derivatives of the function s is negative negative 2 times t to negative first, plus 4 divided by t to the second power.
00:16
Before we actually take the derivative, i want to rewrite this function into powers because i really don't want to have to use quotient rule 4, 4 over t squared.
00:29
So we can first re -break this as negative to t to the negative first power plus four times minus, or four times t to the negative two.
00:42
Now, to take the derivative of each of these will be powerful.
00:47
So we want to find first ds by dt.
00:52
Well, this is going to be ddt of minus 2 t to the negative.
01:02
To first plus 4 t to the negative second power.
01:07
We can use the sum and scalar properties of derivative to rewrite this as negative 2 d d t to the negative first power plus 4 d d t to the negative second power.
01:29
And so this is by sum slash constant property.
01:36
And now we can apply power rule for each of these.
01:42
So we'll be negative 2.
01:43
So we pull negative 1 out front and then subtract 1 off of that.
01:48
So it'll be negative 1 multiplied by negative 2 now, since we have to subtract off 1 from it.
01:59
Then we add to this the derivative of t squared, or negative t squared.
02:07
So we move that out front, subtract 1 off it.
02:11
So it'll be negative 2 times t to the negative third power.
02:18
Now let's just go ahead and simplify this a little bit.
02:21
So our negatives here will cancel out.
02:24
So this will be 2 t to the negative second...