00:01
Right, so in this problem we are given the following function.
00:03
Notice that in this problem we have f as a function of t.
00:09
So f of t is equal to t times sine of pi t.
00:16
So to find the derivative of this, we're going to be using two rules.
00:20
Primarily, we're going to be using the product rule.
00:22
So let us write this on the right side here.
00:25
So the product rule is when we're trying to find the derivative of some function.
00:34
Let's call this function u times v prime.
00:39
The derivative of u times v is going to be u prime times v plus v prime times u.
00:45
In other words, the derivative of the first, keep the second, plus the derivative of the second, and then keep the first.
00:51
That's the first thing.
00:52
Number two, we're also going to be using a second rule.
00:56
This is a chain rule applied on sine pi t.
01:00
Note, the derivative with respect to t of sine of some constant, let's call this constant k times t.
01:11
The derivative for this, or instead of saying kt, let's use u as an example, applying the chain rule on sine of u.
01:20
The derivative for this will be the derivative of sine, which is cosine of u...