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Main content
Current time:0:00Total duration:4:24
AP.CALC:
FUN‑3 (EU)
,
FUN‑3.A (LO)
,
FUN‑3.A.1 (EK)

Video transcript

what we're going to do in this video is get some practice taking derivatives with the power rule so let's say when you take the derivative with respect to X of 1 over X what is that going to be equal to pause this video and try to figure it out so at first you might say how does the power rule apply here for the power rule just to remind ourselves it tells us that if we're taking the derivative of X to the n with respect to X so for taking the derivative of that that that's going to be equal to we take the exponent bring it out front and we've proven it in other videos but this is going to be n times X to the and then we decrement the exponent so n minus 1 but this does not look like that and the key is to appreciate that 1 over X is the same thing as X to the negative 1 so this is going to be the derivative with respect to X of X to the negative 1 and now this looks a lot more like what you might be used to where this is going to be equal to you take our exponent bring it out front so that's negative 1 times X to the negative 1 minus 1 negative 1 minus 1 and so this is going to be equal to negative X to the negative 2 and we're done let's do another example let's say that we're told that f of X is equal to the cube root of x and we want to figure out what f prime of X is equal to pause the video and see if you can figure it out again well once again you might say hey how do I take the derivative of something like this especially if my goal or if I'm thinking that maybe the power rule might be useful and the idea is to rewrite this as an exponent if you could rewrite the cube root as X to the 1/3 power and so the derivative you take the 1/3 bring it out front so it's 1/3 X to the 1/3 minus 1 power and so this is going to be 1/3 times X to the 1/3 minus 1 is negative 2/3 negative 2/3 power and we are done and hopefully through these examples you're seeing that the power rule is incredibly powerful you can you can tackle a far broader range of derivatives than you might have initially thought let's do another example and I'll make this one really nice and hairy let's say we want to figure out the derivative with respect to X of the cube root of x squared what is this going to be and actually let's just not figure out what the derivative is let's figure out the derivative at x equals 8 pause this video again and see if you can figure that out well what we're going to do is first just figure out what this is and then we're going to evaluate it at x equals 8 and the key thing to appreciate is this is the same thing and we're just going to do what we did up here as the derivative with respect to X instead of saying the cube root of x squared we could say this is x squared to the 1/3 power which is the same thing as the derivative with respect to X of well x squared if I raise something to an exponent and then raise that to an exponent I could just take the product of the exponents and so this is going to be X to the 2 times 1/3 power or to the 2/3 power and now this is just going to be equal to I'll do it right over here bring the 2/3 out front 2/3 times X to the what's 2/3 minus 1 well that's 2/3 minus 3/3 or it would be negative 1/3 power and we want to know what happens at x equals 8 so let's just evaluate that that's going to be 2/3 times X is equal to 8 to the negative 1/3 power well what's 8 to the 1/3 power 8 to the 1/3 power is going to be equal to 2 and so 8 to the negative 1/3 power is 1/2 actually let me just do that step-by-step so this is going to be equal to 2/3 times we could do this a 1 over 8 to the 1/3 power and so this is just 1 over 2 2/3 times 1/2 well the that's just going to be equal to 1/3 and we're done
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