First example we have y equals to 3x minus 1 to the 5.
The first thing that we need to do now because we know so many different ways of differentiating,
and because we are dealing with so many several types of mathematical functions,
the first thing we have a moment to just look at this
and to recognize what kind of problem this is
and what rule we should be using so when you look at this closely
you see that you have a large function, something to the power of 5,
and then you have an inside function which is just 3x minus 1.
We can use the chain rule but we can use the faster version of it
so we'll just say dy/dx. You differentiate the outer function first so 5, 3x minus one stays as it is
and decrease the power by 1. So this is just bring the power down,
decrease the power by 1, gentle differentiation and then you multiply it by the differential of the inside
which in our case is just going to be 3. Your final answer for the derivative is 15, 3x minus 1
to the 4 and it doesn?t take as more than a minute to just use the faster version
of the chain rule on questions like this.
Have a look at this second example. Look at it for a moment
and just think what kind of function are you dealing with now
so we have y equals to x squared, 2x plus 1 to the power of 6.
I hope you?ve recognized that there are in fact two functions here.
So you have this function and you have this function which are multiplying together.
You now have to think back and try and remember what you have to do
in order to differentiate two functions that are multiplying together.
So have a little moment and think about what you?ve had to do
in order to differentiate a function like this.
I hope you?ve recalled that in order to differentiate two functions
that are multiplying together, you will have to use the product rule
because these are two product functions, so you then ask yourself:
what is the product rule? You can make a little note of it for yourself
so you can say that dy/dx is just vdudx plus udvdx
or however it is that you?ve learned to and we then apply that straight to this question.
So work ahead of me if you?d like whilst I just to this example for you.
So we will say u is equal to x squared and v equals to 2x plus 1 to the power of 6.
Remember that we now need to differentiate both of them separately
and then we're going to put them into our rule.
So you u dash is equal to 2x, v dash have a look at this,
this is the chain rule so in order to apply the chain rule,
you bring the power down, 2x plus 1 to the 5 and you multiply this with 2
which is the differential of the inside, we now tidy this up to get 12,
2x plus 1 to the power of 5. It?s all done now.
All we have to do is put it into our product rules so we can say that dy/dx is vdudx.
You can draw a little arrows if that helps, and so that you don?t get confused
with the terms so you multiply v first, well, in the product rule it doesn?t matter
what you do first but just to stay consistent with the quotient rule,
you?ll find me always starting with the v.
So you multiply that with the differential of u so we have 2x plus 1 to the power of 6,
multiplied it by 2x and then a plus in the middle and you're now multiplying u with vdudx.
So we have 12, 2x plus 1 to the 5 multiplied by x squared.
We can tidy this up so we've got 2x, plus 1 to the 6.
I could, should put the 2x at the front just to stay consistent
so 2x here and then I have the 12 and the x squared can add together.
It?s not an equals so plus 12x squared, 2x plus 1 to the 5.
There?s more you can do with this obviously, you can simplify this.
We looked at an example where you had common factors
and you can see that we've got a common factor of 2x plus 1 and 2x plus 1 to the power of 5
which is the common factor, so you can take out 2x, plus 1 to the power of 5 out of both terms.
This now leaves you with 2x, 2x plus 1 and you?ve also got 12x squared.
Tidy this up a little bit more so I can see that when I multiply these through
I will get 4x squared plus 2x, plus 12x squared and the only reason I multiply it through
whereas usually I prefer to just leave it factorized,
it's because I can add those two terms together.
So I got 2x plus 1 to the 4, we have 16x squared plus 2x,
and by using the product rule, we have found the derivative of two products
x squared multiplied by 2x plus 1 to the power of 6.
Let?s look at our last example. Observe for a second what?s happening in this function.
We have y equals to 5x minus 1 divided by 3x minus 2.
There should be lots of rules and lots of different functions going through your minds right now
and what rule would be appropriate to differentiating this.
I'll write it down while you have a little think of what kind of function this is.
So we have y equals to 5x minus 1 over 3x minus 2.
You can see, I hope that this is a function that's being divided by a function,
so you have a top function and you have a bottom function.
You also know a method to differentiate this. So think through all the rules that we?ve done
and see if you can remember how to differentiate two functions when they are dividing each other.
I hope you?ve recalled that in order to differentiate two functions that are dividing
you will have to use the quotient rule. Which states vdudx minus udvdx over v squared.
With practice I hope you will actually manage to learn these rules.
But you must continue to practice in order to get good and to get faster at applying these rules.
Okay, so looking at the rule that does mean that we need a u and a v
and the differential of them. Remember what I said, always take the top function as your u
and your bottom function as your v and we're going to differentiate both of them individually.
So u equals to 5x minus 1 and v equals to 3x minus 2, like we do in the product rule,
we differentiate each one of them separately so du/dx,
so u dash in this case just 5, nice and easy and v dash is going to be 3.
Luckily, there was no chain rule this time, it doesn?t happen very often,
so let?s be happy about that. Right, let?s apply dy/dx.
Remember what the rule says vdudx, so we're going to multiply that with that
and then we're going to multiply u with dv/dx remembering this time
that there is a minus in between which is important
and then we're also going to divide it by v squared.
So if I write the 5 first, and then I have 3x minus 2, subtract 3, 5x minus 1
and then all over v squared which is 3x minus 2 squared.
I will leave the 3x minus 2 at the bottom factorized so rather than expanding out just ?cause it?s tidier.
At the top I should be able to simplify some because they're all to the power of 1.
The x's, and we've got some numbers multiply that 2, that gives me 15x minus 10
and that gives me another 15x plus 3, when I multiply that through,
so remember that you?re multiplying it through with the negative 3 so minus 3,
multiplied by a minus 1 will give you a plus 3. We 15x positive and minus 15x
which cancels out leaving us with just minus 10 plus 3 which gives me minus 7 over 3x
minus 2, all squared and if I could see that things would simplify further,
I could?ve expanded the denominator but because nothing else is going to cancel,
we can just think one step ahead of ourselves and leave it as it is.
So we have used the quotient rule to derive or to come to the derivative of y equals to 5x minus 1
divided by 3x minus 2 giving us an answer of minus 7 over 3x minus 2, all squared.
Have a look at the exercise calculations now there's quite few different questions to try out.
All of them will be using a variety of different rules.
Make sure you look at each question first, decide what kind of function that is
and then go for the rule that would suit your function best.
Remember for each different type of function
we have a different rule so make sure you apply it correctly and good luck.