00:01
In your question, we want to find the second derivative of this function and then find the value of the second derivative for three different x values.
00:09
So the first thing we need to understand is the derivative of natural log of u because our inside function here is a function of its own.
00:23
Known.
00:24
The derivative here would be 1 over u times du, the derivative of the inside function.
00:32
That's a chain rule.
00:34
So for first derivative, we would just write 1 over 3x plus 1 times the derivative of 3x plus 1, which is 3.
00:45
So our first derivative is 3 over 3x plus 1.
00:51
Now we want the second derivative, so what i like to do here in this case, instead of using the quotient rule, which is what i would have to use currently, i can rewrite this function as 3 times 3x plus 1 to the negative 1 power.
01:09
That'll make it easier to take our second derivative.
01:13
So for the second derivative, we can use the power rule and the chain rule.
01:19
So the power our rule is going to bring the negative 1 and multiply it to the 3, giving me a negative 3.
01:25
We keep the inside function the same, and we drop the power by 1, taking it to a negative second power.
01:34
We then multiply by the derivative of the inside function, which is 3...