So we've looked at the rules of logs. We'll now going back to differentiating,
so rules of logs are important for us when we need to use them.
We'll see as they come up in different questions.
But if we start to look at differentiating ln's and we'll start with an example
and then we?ll build up from there. So the first question here is asking us to differentiate
ln of 5x cubed plus 7x minus 11. We've been ask to differentiate this
so we'll have to remind ourselves how to differentiate ln of x.
So just on the side, hopefully this is going to our minds that to differentiate ln of x, dy/dx,
that gives you an answer of 1/x. And remember that this is ln of x, gives you 1/x.
Okay, so if it's ln of something else, you'll get 1 over something else.
Let's look at our function for a moment. We have ln, but it's not ln of x,
it's ln of something on the inside. So the first thing that we need to acknowledge
is this is a function inside of a function. We have this function here, which is inside of an ln function,
so we're going to apply the chain rule.
Step by step, we do dy/dx, the first thing that we need to do is just differentiate ln.
So ln of anything is just 1 over that thing.
So because this is unjust, you can treat this as your x.
So ln of x, but instead of ln of x, we've got ln of 5x cubed, plus 7x minus 11.
So we can say 1 over 5x cubed plus 7x minus 11.
So ln of 5x cubed, plus 7x minus 11, differentiates to 1 over exactly the same thing without changing it.
And then remember that we need to multiply it with the differential of the inside.
So what is the differential of this term here, of the inside or this term here, exactly the same.
So the differential of this is going to be 15x squared plus 7.
So your final answer is 15x squared plus 7 that can just come above the fraction,
over 5x cubed plus 7x minus 11. You'll get better at this the more practice you do,
it's quite important to learn to differentiate ln's because remember lots of logs, lots of ln's
and in medicine or understanding pH scales.
So we'll do some practice in the exercise lecture in a minute.
I'll leave you for a while to practice this questions and the exercise calculations
and then I'll meet once you've tried them and then we'll go through the answers.