00:02
They ask us to take the derivative of y equals x -quib or x -cube plus 4x plus 1 all to the 100th power.
00:14
And what we can do then is i can pick this thing in parentheses here and define it as a new variable.
00:22
So we'll call that one x -cubed plus 4x plus 1.
00:26
So then y equals u to the 100th power.
00:31
And so from the chain rule, well, we can take d.
00:35
D -y -u, also in this case here, and that's just 100 times u to the 99th power.
00:43
And then d -y -d -x is d -y -d -u times d -u -d -x.
00:47
So d -y -d -u is 100 times u to the 99th power, but then we'll plug that back into here.
00:52
So we get 100 times the quantity x -cube plus 4x plus 1 to the 99th power.
00:59
And then we need to multiply it on d -u -d -x, and the u -d -x is simply 3x.
01:06
Squared plus four.
01:08
So this is our expression then.
01:11
So even though you have huge power here, it doesn't really matter.
01:16
In fact, you know, if this was a million or which was two, you know, the, the math is going to be just as simple if you use the chain rule.
01:29
And next one they have, if they want us to find a derivative of this function of x, e to the three x, plus 5.
01:39
And so we can write f as a function of u, e to the u of x, where u of x equals 3x squared plus 5.
01:48
And then our chain rule...