Learn about Derivatives - part 8 Video

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Learn about Derivatives - part 8

Welcome back! Let’s do some derivative problems. Let’s say I want to figure out, let’s say I’m going to do the derivative D/Dx of—let me give something that looks a little bit different. (X3– 5X5)3/ (2X+55), this just saying that I want to take the derivative of this entire expression. So you're saying “Sal, we’ve never learned how to do this, you have something the numerator, you have something the denominator, I don’t know what to do next”. Well let’s just rewrite this. They’re actually in your calculus textbooks, they’re 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 derivative of (X3-5X5)3 times (2X+5)-5, right? Then 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 as to the chain rule. The derivative of the inside first that is (3X2-25X4)3 times the derivative of the outside 3 times, this entire expression, (X3 –5X5) and then all of that take this exponent down one to the squared and then multiply it times this whole term. So (2x+ 5)-5 and then to that we add the derivative of this term, so plus the derivative of this term, we take the derivative of inside which is pretty easy is just 2 times the derivative of the outside which is -5. And just so—you know I didn’t skip a step, the derivative of the 2X+ 5 the derivative of 2X is two, the derivative 5 is 0. The derivative of 2X+5 is 2. So it’s 2 times -5(2x+5)-5 and we multiply it times its first expression (X3-5x5)3. I know that’s really messing you, you’re probably not see problems this mess. But I just want to show that the product rule we learned is actually the product and the chain rule that can apply to a lot of different problems and even if you hadn’t seen something like this where it had a numerator and a denominator, you can easily rewrite what you had the denominator as a negative exponent and then of course it’s just the product and you don’t have to memorize that silly thing called the quotient rule. So with that out of the way, I’m not going to introduce you to some common derivatives of other functions and these things are actually normally included in the inside cover of calculus book and it’s just good to know—good, good things to know and maybe in a later presentation I’ll actually prove these things. You should never take things at place value. So these are—you should to some degree memorize these although you should prove it to yourself first. So the derivative of Ex and I find this to be amazing. E shows up all sorts of crazy places in mathematics and as 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 Ex if I want to figure out the slope of any point along the curve Ex just might blow your mind, I think the more you think about it, the more it will blow your mind is Ex. That’s amazing At any point along the curve Ex, the slope of that point is Ex. So if I said just to hit the point home, I’m diverging a little bit. But if I said F of X= Ex, right? And let’s say F of two= E2, right? And I ask you friend, I don’t know your name. What is this slope of Ex at this point at the point (2, E2)? And you could say, “Sal, the slope of that point is E2. That blows my mind. That is 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 place—I might do whole series of presentations where called the magic of E because E shows up all over the place. I don’t know if you—well, I don’t want to diverge too much. So that’s pretty amazing. Next, I’m going to show what I think is probably the second most amazing of derivative and I don’t think 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 the 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. What’s 1/x? That also blows my mind, right? Because think about it. Let’s draw a bunch of functions. If I said, the derivative of X-3 is -3X-4, right? The derivative of X-2 is -2X-3, right? The derivative of X-1 is -1X-2. The derivative of X0, well this is just one right? The derivative of X0 is just 1, so the derivative is 0. The derivative of X is 1; derivative of X2 is 2X and so on, right? So it’s interesting. We have this pattern, right, from all the derivatives of all the of kind of the exponents and increasing whatever you go from X-4, X-3, X-2, and then there’s no X-1 here, right? And we go straight to X0. Like what happen to—what happen to X1? Right? What happen to this? What functions derivative is X-1? This is bizarre to me. Where did it go? And it turns out that is a natural log. This I still think about before I go to bed sometimes because it is kind of mind blowing. So and later in other presentation I’m actually prove this to you but just to know that this is true, 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 Ex is Ex and derivative of the natural log of X is 1/X. And I’ll just do a couple of more or 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 etcetera, etcetera. You might want to rewatch this and memorize it but let’s see, I want to view the image, image. Okay? And now just do the basic trick functions and these are—you should memorize this as well. The derivative of sine X= cosine X, so the slope at any point along the line sine of X is actually the cosine of that point. That’s also interesting. What I’m going to do is I’ll graph it but I think that might not be sinking in properly. The derivative of cosine of X is –sine of X. These are good to memorize them because you’ll be able to recall it quickly in a test and then use it. And then finally, the derivative of tan x= 1 over cosine square root of X which is you could also write as the secant squared of X. You might want to memorize this 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 probably do presentation on that. But I almost out of time in this presentation, so just memorize these and memorize the derivative Ex and Ex with 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 a product and chain rule them to solve kind of arbitrarily complex derivatives. So that’ll probably between what you’ve just seen, we could probably solve 95% of the derivative problems and we’ll see or I’m saying the calculus AP test. I’ll see you in the presentation.

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Welcome back! Let’s do some derivative problems. Let’s say I want to figure out, let’s say I’m going to do the derivative D/Dx of—let me give something that looks a little bit different. (X3– 5X5)3/ (2X+55), this just saying that I want to take the derivative of this entire expression. So you're saying “Sal, we’ve never learned how to do this, you have something the numerator, you have something the denominator, I don’t know what to do next”.

Well let’s just rewrite this. They’re actually in your calculus textbooks, they’re 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 derivative of (X3-5X5)3 times (2X+5)-5, right? Then 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 as to the chain rule. The derivative of the inside first that is (3X2-25X4)3 times the derivative of the outside 3 times, this entire expression, (X3 –5X5) and then all of that take this exponent down one to the squared and then multiply it times this whole term. So (2x+ 5)-5 and then to that we add the derivative of this term, so plus the derivative of this term, we take the derivative of inside which is pretty easy is just 2 times the derivative of the outside which is -5. And just so—you know I didn’t skip a step, the derivative of the 2X+ 5 the derivative of 2X is two, the derivative 5 is 0. The derivative of 2X+5 is 2.

So it’s 2 times -5(2x+5)-5 and we multiply it times its first expression (X3-5x5)3. I know that’s really messing you, you’re probably not see problems this mess. But I just want to show that the product rule we learned is actually the product and the chain rule that can apply to a lot of different problems and even if you hadn’t seen something like this where it had a numerator and a denominator, you can easily rewrite what you had the denominator as a negative exponent and then of course it’s just the product and you don’t have to memorize that silly thing called the quotient rule.

So with that out of the way, I’m not going to introduce you to some common derivatives of other functions and these things are actually normally included in the inside cover of calculus book and it’s just good to know—good, good things to know and maybe in a later presentation I’ll actually prove these things. You should never take things at place value.

So these are—you should to some degree memorize these although you should prove it to yourself first. So the derivative of Ex and I find this to be amazing. E shows up all sorts of crazy places in mathematics and as 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 Ex if I want to figure out the slope of any point along the curve Ex just might blow your mind, I think the more you think about it, the more it will blow your mind is Ex. That’s amazing

At any point along the curve Ex, the slope of that point is Ex. So if I said just to hit the point home, I’m diverging a little bit. But if I said F of X= Ex, right? And let’s say F of two= E2, right? And I ask you friend, I don’t know your name. What is this slope of Ex at this point at the point (2, E2)? And you could say, “Sal, the slope of that point is E2. That blows my mind. That is 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 place—I might do whole series of presentations where called the magic of E because E shows up all over the place. I don’t know if you—well, I don’t want to diverge too much. So that’s pretty amazing.

Next, I’m going to show what I think is probably the second most amazing of derivative and I don’t think 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 the 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. What’s 1/x? That also blows my mind, right? Because think about it. Let’s draw a bunch of functions. If I said, the derivative of X-3 is -3X-4, right? The derivative of X-2 is -2X-3, right? The derivative of X-1 is -1X-2. The derivative of X0, well this is just one right? The derivative of X0 is just 1, so the derivative is 0. The derivative of X is 1; derivative of X2 is 2X and so on, right?

So it’s interesting. We have this pattern, right, from all the derivatives of all the of kind of the exponents and increasing whatever you go from X-4, X-3, X-2, and then there’s no X-1 here, right? And we go straight to X0. Like what happen to—what happen to X1? Right? What happen to this? What functions derivative is X-1? This is bizarre to me. Where did it go? And it turns out that is a natural log.

This I still think about before I go to bed sometimes because it is kind of mind blowing. So and later in other presentation I’m actually prove this to you but just to know that this is true, 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 Ex is Ex and derivative of the natural log of X is 1/X.

And I’ll just do a couple of more or 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 etcetera, etcetera. You might want to rewatch this and memorize it but let’s see, I want to view the image, image. Okay? And now just do the basic trick functions and these are—you should memorize this as well. The derivative of sine X= cosine X, so the slope at any point along the line sine of X is actually the cosine of that point. That’s also interesting. What I’m going to do is I’ll graph it but I think that might not be sinking in properly.

The derivative of cosine of X is –sine of X. These are good to memorize them because you’ll be able to recall it quickly in a test and then use it. And then finally, the derivative of tan x= 1 over cosine square root of X which is you could also write as the secant squared of X.

You might want to memorize this 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 probably do presentation on that.

But I almost out of time in this presentation, so just memorize these and memorize the derivative Ex and Ex with 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 a product and chain rule them to solve kind of arbitrarily complex derivatives. So that’ll probably between what you’ve just seen, we could probably solve 95% of the derivative problems and we’ll see or I’m saying the calculus AP test. I’ll see you in the presentation.

Transcription by: Scribe4you Transcription Services