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# Basic derivative rules: find the error

AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.2 (EK)

## Video transcript

so we have two examples here someone trying to find the derivative of an expression or on the left hand side it says Avery tried to find the derivative of 7 minus 5x using basic differentiation rules here is her work and on the right hand side it says Hannah tried to find the derivative of negative 3 plus 8x using basic differentiation rules here is her work and these are two different examples frenchie ation rules exercise on Khan Academy and I thought I would just do them side by side because we can kind of think about what each of these people are doing correct or incorrect so these are similar expressions we have a constant and then we have a first degree term a constant and then a first degree term so they're going to take the derivative let's see step one for Avery she took she separately taking the derivative of 7 and separately taking the derivative of 5x so this is my spider-sense is already going off here because what happened to this negative right over here so it would have made sense for her to do the derivative of 7 and she could have said minus the derivative of 5x that's one possibility that she could have done the derivative of a difference is equal to the difference of the derivatives we've seen that property or she could have said the derivative she could have said this is equal to the derivative of 7 plus the derivative with respect to X of negative 5x these two things would have been equivalent to this one but for this one she somehow forgot to include the negative so I think she had a problem right at step one now if you just follow her logic after step one let's see if she makes any more mistakes so she takes the derivative of a constant so a constant isn't going to change with respect to X so that makes sense that that derivative is zero and so we still have the derivative of 5x and remember it should have been negative 5x or minus the derivative of 5x and let's see what she does here so that zero disappears and now she takes the and she takes the constant out and that's true the derivative of a constant time something is equal to the constant times the derivative of that something and then she finds the derivative with respect to X of X is 1 and that's true if the slope if you had the graph of y equals x the slope there's 1 or what's the rate of change at which X changes with respect to X well it's going to be 1 for 1 and so the slope here is 1 so this is going to be 5 times 1 which is equal to 5 and at the end to say what step did Avery make a mistake so she clearly made a mistake at step one this right of here should have been a negative if that's a negative then that would have been a negative then this would have been a negative then that would have been a negative and then her final answer should have been should have been a negative 5 now let's go back to Hannah to see if she made any mistakes and where so she's differentiating a similar expression so first she takes the derivative of the constant plus the derivative of the first degree term derivative of a constant is 0 that looks good so you get the 0 and then you have the derivative of the first degree term that's what she's trying to figure out and then let's see she's taking ok so this is this seems off she is assuming that the derivative of a product is equal to the product of the derivatives that is not the case and especially and it's if you have a constant here there's actually a much simpler way of thinking about it frankly the way that Avery thought about it Avery had made a mistake at step one but this is actually going to be equal to the derivative of a constant times an expression is equal to the same thing as the constant times the derivative of of the expression so this would have been the correct way to go and then the derivative of X with respect to X well that's just going to be 1 so this should have all simplified to 8 what she did is she is assumed she tried to take the derivative of 8 and multiply that times the derivative X that is not the way it works in the future you will learn something called the product rule but you don't even have to apply that here because one of these one of these components I guess you could say is a constant so this is the wrong step this is where Hannah makes a mistake and you can see instead of getting a final answer of eight she's getting a final answer of she assumes well the derivative of eight is zero times the derivative of X is one zero times one and she gets zero which is not the right answer so she makes a mistake at step three and Avery made a mistake at step one
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