Welcome back. Let’s do some more derivative problems.
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Quotient rule and common derivatives Taking derivatives Differential Calculus
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. OK. 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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