Power rule introduction Determining the derivatives of simple polynomials.
Power rule introduction
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- Welcome back.
- In the last presentation I showed you that if I had the
- function f of x is equal to x squared, that the derivative of
- this function, which is denoted by f-- look at that, my pen
- is already malfunctioning.
- The derivative of that function, f prime of
- x, is equal to 2x.
- And I used the limit definition of a derivative.
- I used, let me write it down here.
- This pen is horrible.
- I need to really figure out some other tool to use.
- The limit as h approaches 0 -- sometimes you'll see delta x
- instead of h, but it's the same thing-- of f of x plus
- h minus f of x over h.
- And I used this definition of a derivative, which is really
- just the slope at any given point along the curve,
- to figure this out.
- That if f of x is equal to x squared, that
- the derivative is 2x.
- And you could actually use this to do others.
- And I won't do it now, maybe I'll do it in a
- future presentation.
- But it turns out that if you have f of x is equal to x to
- the third, that the derivative is f prime of x is
- equal to 3x squared.
- If f of x is equal to x to the fourth, well then the
- derivative is equal to 4x to the third.
- I think you're starting to see a pattern here.
- If I actually wrote up here that if f of x -- let me see
- if I have space to write it neatly.
- If I wrote f of x -- I hope you can see this -- f
- of x is equal to x.
- Well you know this.
- I mean, y equals x, what's the slope of y equals x?
- That's just 1, right?
- y equals x, that's a slope of 1.
- You didn't need to know calculus to know that.
- f prime of x is just equal to 1.
- And then you can probably guess what the next one is.
- If f of x is equal to x to the fifth, then the derivative is--
- I think you could guess-- 5 x to the fourth.
- So in general, for any expression within a polynomial,
- or any degree x to whatever power-- let's say f of x is
- equal to-- this pen drives me nuts.
- f of x is equal to x to the n, right?
- Where n could be any exponent.
- Then f prime of x is equal to nx to the n minus 1.
- And you see this is what the case was in all
- these situations.
- That 1 didn't show up.
- n minus 1.
- So if n was 25, x to the 25th power, the derivative
- would be 25 x to the 24th.
- So I'm going to use this rule and then I'm going to show
- you a couple of other ones.
- And then now we can figure out the derivative of pretty much
- any polynomial function.
- So just another couple of rules.
- This might be a little intuitive for you, and if you
- use that limit definition of a derivative, you could
- actually prove it.
- But if I want to figure out the derivative of, let's say, the
- derivative of-- So another way of-- this is kind of, what is
- the change with respect to x?
- This is another notation.
- I think this is what Leibniz uses to figure out the
- derivative operator.
- So if I wanted to find the derivative of A f of x, where A
- is just some constant number.
- It could be 5 times f of x.
- This is the same thing as saying A times the
- derivative of f of x.
- And what does that tell us?
- Well, this tells us that, let's say I had f of x.
- f of x is equal to-- and this only works with the constants--
- f of x is equal to 5x squared.
- Well this is the same thing as 5 times x squared.
- I know I'm stating the obvious.
- So we can just say that the derivative of this is just 5
- times the derivative of x squared.
- So f prime of x is equal to 5 times, and what's the
- derivative of x squared?
- Right, it's 2x.
- So it equals 10x.
- Similarly, let's say I had g of x, just using
- a different letter.
- g of x is equal to-- and my pen keeps malfunctioning.
- g of x is equal to, let's say, 3x to the 12th.
- Then g prime of x, or the derivative of g, is equal
- to 3 times the derivative of x to the 12th.
- Well we know what that is.
- It's 12 x to the 11th.
- Which you would have seen.
- 12x to the 11th.
- This equals 36x to the 11th.
- Pretty straightforward, right?
- You just multiply the constant times whatever the
- derivative would have been.
- I think you get that.
- Now one other thing.
- If I wanted to apply the derivative operator-- let me
- change colors just to mix things up a little bit.
- Let's say if I wanted to apply the derivative of operator-- I
- think this is called the addition rule.
- It might be a little bit obvious.
- f of x plus g of x.
- This is the same thing as the derivative of f of x plus
- the derivative3 of g of x.
- That might seem a little complicated to you, but all
- it's saying is that you can find the derivative of each of
- the parts when you're adding up, and then that's the
- derivative of the whole thing.
- I'll do a couple of examples.
- So what does this tell us?
- This is also the same thing, of course.
- This is, I believe, Leibniz's notation.
- And then Lagrange's notation is-- of course these were the
- founding fathers of calculus.
- That's the same thing as f prime of x plus g prime of x.
- And let me apply this, because whenever you apply it, I think
- it starts to seem a lot more obvious.
- So let's say f of x is equal to 3x squared plus 5x plus 3.
- Well, if we just want to figure out the derivative, we say f
- prime of x, we just find the derivative of each
- of these terms.
- Well, this is 3 times the derivative of x squared.
- The derivative of x squared, we already figured
- out, is 2x, right?
- So this becomes 6x.
- Really you just take the 2, multiply it by the 3, and
- then decrement the 2 by 1.
- So it's really 6x to the first, which is the same thing as 6x.
- Plus the derivative of 5x is 5.
- And you know that because if I just had a line that's y equals
- 5x, the slope is 5, right?
- Plus, what's the derivative of a constant function?
- What's the derivative of 3?
- Well, I'll give you a hint.
- Graph y equals 3 and tell me what the slope is.
- Right, the derivative of a constant is 0.
- I'll show other times why that might be more intuitive.
- Plus 0.
- You can just ignore that.
- f prime of x is equal to 6x plus 5.
- Let's do some more.
- I think the more examples we do, the better.
- And I want to keep switching notations, so you don't get
- daunted whenever you see it in a different way.
- Let's say y equals 10x to the fifth minus 7x to the
- third plus 4x plus 1.
- So here we're going to apply the derivative operator.
- So we say dy-- this is I think Leibniz's
- notation-- dy over dx.
- And that's just the change in y over the change in x,
- over very small changes.
- That's kind of how I view this d, like a very small delta.
- Is equal to 5 times 10 is 50 x to the fourth minus 21 --
- right, 3 times 7-- x squared plus 4.
- And then the 1, the derivative of 1 is just 0.
- So there it is.
- We figured out the derivative of this very
- complicated function.
- And it was pretty straightforward.
- I think you'll find that derivatives of polynomials are
- actually more straightforward than a lot of concepts that you
- learned a lot earlier in mathematics.
- That's all the time I have now for this presentation.
- In the next couple I'll just do a bunch of more examples, and
- I'll show you some more rules for solving even more
- complicated derivatives.
- See you in the next presentation.
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