Quotient rule and common derivatives Why the quotient rule is the same thing as the product rule. Introduction to the derivative of e^x, ln x, sin x, cos x, and tan x
Quotient rule and common derivatives
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- Welcome back.
- Let's do some more derivative problems.
- Let's say I want to figure out the derivative d over dx of--
- and let me give something that looks a little bit different--
- x to the third minus 5x to the fifth, all of that to the third
- power over 2x plus 5 to the fifth power.
- This is a parentheses.
- This is just saying that I want to take the derivative of
- this entire expression.
- So you're saying Sal, we've never learn how to do this,
- you have something in the numerator, you have something
- in the denominator I don't know what to do next.
- Well let's just rewrite this.
- Actually in your calculus textbooks there's something
- called the quotient rule, which I think is mildly lame, because
- the quotient rule is just the product rule where you have a
- negative exponent and they make it another rule, and they
- clutter your brain.
- So instead of using the quotient rule, we're just
- going to rewrite this bottom expression as a product,
- and then we can use the product rule.
- So this is the same thing as taking the derivative of x to
- the third minus 5x to the fifth, all of that to the third
- power, times 2x plus 5 to what?
- The minus fifth power.
- And now we can use the product rule.
- Take the derivative of the first term-- and the derivative
- of the first term isn't a joke-- you take the derivative
- of the inside first, let's do the chain rule, derivative
- of the inside first.
- That is 3x squared minus 25x to the fourth times the derivative
- of the outside, 3 times this entire expression x to the
- third minus 5x to the fifth.
- And then all of that, take this exponent down one to the
- squared, and then multiply it times this whole term.
- So 2x plus 5 to the minus fifth.
- And then to that we add the derivative of
- this term, so plus.
- So the derivative of this term we take the derivative of the
- inside, which is pretty easy.
- It's just 2 times the derivative of the outside,
- which is minus 5.
- And just so you know I didn't skip a step, the derivative of
- 2x plus 5, the derivative of 2x is 2, derivative of 5 is 0.
- So the derivative of 2x plus 5 is just 2.
- So it's 2 times minus 5 2x plus 5.
- We just keep that the same to the minus fifth power, and then
- we multiply it times this first expression, x to the third
- minus 5x to the fifth to the third power.
- I know that's really messy and you'll probably not see
- problems this messy, but I just wanted to show you that the
- product rule we learned-- it's actually the product and the
- chain rule-- they can apply to a lot of different problems,
- and even though you hadn't seen something like this where you
- had numerator and a denominator, you can easily
- rewrite what you had in the denominator as a
- negative exponent.
- And then of course it's just the product for when you don't
- have to memorize that silly thing called the quotient rule.
- So with that out of the way, I'm now going to introduce you
- to some common derivatives of other functions.
- And these things are actually normally included in the inside
- cover of your calculus book, and they're just good to
- know, good things to know.
- And maybe in a later presentation I'll actually
- prove these things.
- You should never take things at face value.
- So you should to some degree memorize these, although you
- should prove it to yourself first.
- So the derivative of e to the x-- and I find this to be
- amazing. e shows up all sorts of crazy places in mathematics,
- and it's you know the strange number 2.7 whatever, whatever
- and it has all sorts of strange properties.
- And I think this is one of the most bizarre properties of e.
- The derivative of e to the x.
- So if I want to figure out the slope of any point along
- the curve e to the x-- this just might blow your mind.
- I think the more you think about it, the more it'll blow
- your mind-- is e to the x.
- That's amazing.
- At any point along the curve e to the x, the slope of
- that point is e to the x.
- Just to hit the point home.
- I'm diverging, a little bit.
- But if I said f of x is equal to e to the x, right?
- And let's say f of 2 is equal to e squared.
- And I asked you, friend-- I don't know your name-- what
- is the slope of e to the x at the point 2,e squared.
- And you could say Sal, the slope at that
- point is e squared.
- That blows my mind that it's a function where the slope at
- any point on that line is equal to the function.
- And it's e. e shows up all sorts of places.
- I might do a whole series of presentations called the
- magic of e, because e shows up all over the place.
- Well I don't want to diverge too much, so that's
- pretty amazing.
- Next I'm going to show you what I think is probably the second
- most amazing derivative-- and I don't think this has been fully
- explored in mathematics yet, because this also blows my
- mind-- is that the derivative of the natural log of x, right.
- So the natural log is just the logarithm with base e, and I
- hope you remember your logarithms.
- So what's the derivative of the natural log of x?
- So once again this is e related.
- Well it's 1/x.
- That also blows my mind.
- Because think about it.
- Let's draw a bunch of functions.
- If I said the derivative of x to the minus 3 is
- minus 3x to the minus 4.
- The derivative of x to the minus 2 is minus
- 2x to the minus 3.
- The derivative of x to the minus 1 is minus
- 1 x to the minus 2.
- The derivative of x to the 0-- well this is just 1, right?
- The derivative of x to the 0 is just 1, so the derivative is 0.
- The derivative of x is 1, derivative of x squared
- is 2x and so on, right?
- So it's interesting.
- We have this pattern from all the derivatives of all of the
- of kind of the exponents in increasing order where you go
- from x to the minus 4 x to the minus 3, x to the minus 2 and
- then there's no x to the minus 1 here.
- We go straight to x to the 0.
- What happened to x the minus 1?
- What happened to this?
- What function's derivative is x to the minus 1?
- This is bizarre to me.
- Where did it go?
- And it turns out that it's a natural log.
- This I still think about before I go to bed sometimes because
- it is kind of mind blowing.
- And later in another presentation I might
- actually prove this to you.
- But just to know that this is true, that the derivative of
- the natural log of x is 1/x I think is mind blowing.
- And so for now you can just memorize it.
- But both of these are mind blowing.
- The derivative of e to the x is e to the x, and the derivative
- of the natural log of x is 1/x.
- And I'll just do a couple of more just to present them to
- you, and then in the next presentation we'll actually use
- them using the product rule and the chain rule and et
- cetera, et cetera.
- And you might want to rewatch this and memorize them.
- I want to clear image.
- And now I'll just do the basic trig functions, and you should
- memorize these as well.
- The derivative of sin of x-- this is pretty easy to
- remember-- is cosine of x.
- So the slope at any point along the [? line ?]
- sin of x is actually the cosine of that point.
- That's also interesting.
- One day I'm going to do this holographically because I
- think that might not be sinking in properly.
- The derivative of cosine of x is minus sin of x.
- There are good to memorize though, because you'll be
- able to recall is quickly on a test and then use it.
- And then finally the derivative of tan of x is equal to 1 over
- cosine square of x which you could also write as the
- secant squared of x.
- You might want to memorize these now, and actually I
- encourage you to explore these things, I encourage you to
- graph each of these functions.
- Graph a function, graph its derivative and look at them,
- and really intuitively understand why the derivative
- function actually does describe the slope of
- the original function.
- And actually I'll probably do a presentation on that.
- But I'm almost out of time in this presentation,
- so just memorize these.
- And memorize the derivative of e to the x, e to the x, and
- the natural log of x is 1/x.
- And in the next presentation we're going to start mixing and
- matching all of these functions, and we can use the
- product and chain rule on them to solve kind of arbitrarily
- complex derivatives.
- Between what we've just seen, we could probably solve 95% of
- the derivative problems you'll see on say the
- calculus AP test.
- I'll see you in the next presentation.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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