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Current time:0:00Total duration:9:53

AP.CALC:

FUN‑3 (EU)

, FUN‑3.A (LO)

, FUN‑3.A.2 (EK)

now that we know the power rule and we saw that in the last video that the derivative with respect to X of X to the N is going to be equal to n times X to the n minus 1 for n not equal 0 I thought I would expose you to a few more rules or concepts or properties of derivatives it essentially will allow us to take the derivative of any polynomial so this is powerful stuff going on so the first thing I want to think about is why this little special case for n not equaling 0 what happens if N equals 0 so let's just think of the situation let's try to take the derivative with respect to X of X to the 0 power well what is X to the 0 power going to be well and we can assume that X for this case right over here is not equal to 0 0 to the 0 weird things happen at that point but if X does not equal 0 what is X to the 0 power going to be well this is the same thing as the derivative with respect to X of 1 X to the 0 power is just going to be 1 and so what is the derivative with respect to what with the derivative with respect to X of 1 and to answer that question I'll just graph it I'll just graph f of X equals 1 to make it a little bit clearer so that's my y-axis this is my x-axis and let me graph y equals 1 or f of x equals 1 so that's 1 right over there F of x equals 1 it's just a horizontal line so that right over there is the graph y is equal to f of X which is equal to 1 now remember the derivative one way to conceptualize it is just the slope of the tangent line at any point so what is the slope of the tangent line at this point and actually what's the slope at every point well this is a line so the slope doesn't change has a constant slope and it's completely horizontal line it has a slope of 0 so the slope at every point over here slope is going to be equal to 0 so the slope of the the slope of this line at any point is just going to be equal to 0 and that's actually going to be true for any constant the derivative if I had a function ad f of X is equal to 3 let's say that that's Y is equal to 3 what's the derivative of Y with respect to X going to be equal to and I'm intentionally changed showing you all the different ways of the notation for derivative so what's the derivative of Y with respect to X it can also be written as Y prime what's that going to be equal to well it's the slope at any given point and you see that no matter what X you're looking at the slope here is going to be 0 so it's going to be 0 so it's not just X to the 0 if you take the derivative of any constant you're going to get 0 so let me write that derivative with respect to X of any constant so let's say that of a where this is just a constant where this is just going to be a constant that's going to be equal to 0 so pretty straightforward idea now let's explore a few more properties let's say our 2t I want to take the derivative with respect to X of let's use the same a let's say have some constant times some function times some function well derivatives work out quite well you can actually take this this little scalar multiplier this little constant and take it out of the derivative this is going to be equal to a it's going to be equal to a I don't want to do that in magenta color it's going to be equal to a times the derivative of f of X a times the derivative of f of X the blue color of f of X and the other way to denote the derivative of f of X is to just say that this is the same thing this is equal to a times this thing right over here is the exact same thing as f prime of X now this might all look like really fancy notation but I think if I gave you an example it might make some sense so what about if I were to ask you the derivative with respect to X the derivative with respect to X of 2 of 2 x times X to the fifth power well this property that I just articulated says well this is going to be the same thing as two times the derivative this is going to be the same thing as two times the derivative of X to the fifth two times the derivative with respect to X of X to the fifth essentially I could just take this scalar multiplier and put it in front of the derivative so this right here this is the derivative with respect to X of X to the fifth and we know how to do that using the power rule this is going to be equal to two times let me write that I want to keep it consistent with the colors this is going to be two times derivative of X to the fifth well the power rule tells us n is five is going to be 5 X to the 5 minus 1 or 5 X to the fourth power so it's going to be 5 X to the fourth power which is going to be equal to 2 times 5 is 10 X to the fourth so 2 X to the fifth you can literally just say okay the power rule tells me the derivative of that is 5x to the fourth 5 times 2 is 10 so that simplifies our life a good bit we can now using the power rule and this one property take the derivative of anything that takes the form ax to the N power now let's think about another very useful derivative property and these don't just apply to the power rule they apply to any derivative but they are especially useful for the power rule because it allows us to construct polynomials and take the derivatives of them but if I were to take the derivative of the sum of two functions so the derivative of let's say one function is f of X and then the other function is G of X it's lucky for us that this ends up being the same thing as the derivative of f of X plus the derivative of G of X so this is the same thing as F actually let me use that derivative operator just to make it clear it's the same thing as the derivative with respect to X of f of X plus the derivative with respect to X of G of X so we put f of X right over here and put G of X put G of X right over there and so with the other notation we could say this is going to be the same thing derivative with respect to X of f of X we can write as f prime of X and the derivative with respect to X of G of X we can write as G prime of X now once again this might look like kind of fancy notation to you but when you see an example it'll make it pretty clear if I have if I want to take the derivative with it with respect to X of let's say X to the third power plus X to the negative 4 power this just tells us that the derivative of the sum is just the sum of the derivatives so we can take the derivative of this term using the power rule so it's going to be 3x squared + 2 that we can add the derivative of this thing right over here so it's going to be + that's a different shade of blue + and over here is negative 4 so it's plus negative 4 plus negative 4 times X to the negative 4 minus 1 or X to the negative 5 power so we have and I can just simplify a little bit this is going to be equal to 3 this is going to be equal to 3x squared - 4 X to the negative 5 and so now we have all the tools we need in our toolkit to essentially take the derivative of any polynomials so let's give ourselves a little practice there so let's say that I have let's say that I have and I'll do it in a I'll do it in white let's say that f of X is equal to 2 X to the third power minus 7x squared plus 3x minus 100 what is f prime of X what is the derivative of F with respect to X going to be well we can use the properties that we just said the derivative of this is just going to be 2 times the derivative of X to the third derivative of X to the third is going to be 3x squared so it's going to be 2 times 3x squared what's the derivative of negative 7x squared going to be what's going to be negative seven times the derivative of x squared which is 2x 2x what is the derivative of 3x going to be what's just going to be 3 times the derivative of X or 3 times the derivative of X to the first the derivative of X to the first is just 1 so this is just going to be plus 3 times we could say 1 X to the 0 but that's just 1 and then finally what's the derivative of a constant going to be let me do that in a different color what's the derivative of a constant going to be well we cover that at the beginning of this video the derivative of any constant is just going to be 0 so plus 0 and so now we are ready to simplify the derivative is of F is going to be 2 times 3 x squared is just 6 x squared negative 7 times 2x is negative 14x plus 3 and we don't have to write the zero there and we're done we now have all the properties in our tool belt to find the derivative of any polynomial and actually things that even go beyond polynomials