We have a much faster way of doing this which you all, will probably be familiar with already
and that states that if you have a function y equals to x to the power of n,
the differential of that which is probably been derived
just by observation and by looking and lots and lots and lots of different types of gradients.
The differential of that is given by bringing the power n to the front.
So you bring it down as a multiple.
So you can see here that if its y equals to x to the power of n,
you bring n down as a multiple and then the last step is that you decrease the power by 1.
So you have n times x to the power of n minus 1.
This is just the general definition but you'll see that as we do this numerically it makes a lot more sense.
So we spent a whole and 5 or so minutes deriving the differential of 3x squared plus 5x plus 1.
But if you apply to this method by just bringing the power down and decrease the power by 1.
You can see that you'll reach to, reach their answer of 6x plus 5 a lot faster.
So we've just derived the derivative for differentiation for first principles
and we've used on numerical examples.
We've also like at a faster way of using differentiation,
now it's your turn so have a go at these three questions
and try them with using the definition for differentiation from first principles
but then also try the faster way just to check
because you'll be able to know the answer before you actually derive your answer by first principles.
I'm just going to say that the first and the last one are a little bit of a challenge
so you will find that you may have to use binomial theorem to expand it.
But good luck and I'll see you in the exercise lecture.