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Current time:0:00Total duration:4:49

Differentiating integer powers (mixed positive and negative)

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

Video transcript

so we have the function G of X which is equal to 2 over X to the third minus 1 over x squared and what I want to do in this video is I want to find I want to figure out what G prime of X is and then I also want to evaluate that at x equals 2 so I want to figure that out and I also want to figure out what does that evaluate to when X is equal to 2 so what is the slope of the tangent line to the graph of G when X is equal to 2 and like always pause this video and see if you can work this out on your own before I work through it with you and I'll give you some hints all you really need to do is apply the power rule a little bit of basic exponent properties and some basic derivative properties to be able to do this all right now let's just do this together and I'll just rewrite it G of X is equal to this first term here 2 over X to the third well that could be re-written as 2 times X to the negative 3 we know that 1 over X to the N is the same thing as X to the negative n so I just rewrote it and now this might be ringing a bell of how the power rule might be useful and then we have minus well 1 over x squared that is the same thing as X to the negative 2 and so this if we're going to take the derivative of both sides of this let's do that derivative with respect to X DX we're going to do that on the left hand side we're also going to do it on the right hand side on the left hand side the derivative with respect to X of G of X we can write that as G prime of X is going to be equal to well the derivative of this first that what we have right here written in green this is going to be we're just going to apply the power rule we're going to take our exponent multiply it by our coefficient out front actually let me write that out so that's going to be me finish this equal sign that is going to be 2 times negative 3 times X and now we're going to decrement this exponent you have to be very careful here because sometimes your brain might say ok 1 less than 3 is 2 so maybe this is X to the negative 2 but remember you're going down so if you're at negative 3 and you subtract 1 we're going to go to the negative 3 minus 1 power well that's going to take us to negative 4 so this is X to the negative 4 power so 2 times negative 3 X to the negative 4 or we could have also written that as negative 6 X to the negative 4 power and then - well we're going to do the same thing again all right over here we take this negative 2 multiply it times the coefficient that's implicitly here you could say there's a 1 there so negative 2 times 1 so you have the negative 2 there and then you have the X to the well what's negative 2 minus 1 well that's negative 3 to the negative 3 power and so we can rewrite all of this business as the derivative G prime of X is equal to negative 6 negative 6 X to the negative 4th and now we're subtracting a negative so we could just write this as plus 2 X to the negative 3 this negative cancels out with that negative subtract a negative the same thing as adding the positive so we did the first part we we can express G prime of X as a function of X now let's just evaluate what G prime of 2 is so G prime of 2 is going to be equal to negative 6 times 2 to the negative fourth power plus 2 times 2 to the negative third power well what's this going to be this is equal to negative 6 over 2 to the 4th plus 2 over 2 to the 3rd which is equal to negative 6 over 2 to the 4th is 16 plus 2 over 2 to the 3rd is 8 and so let's see this is we could rewrite this let's itright all of this with a common denominator I could write this as 1/4 but then this one won't work out as cleanly I could write them both as ates this is negative 3/8 negative 3/8 so you have negative 3/8 plus two eighths is equal to negative 1/8 so the slope of the tangent line at x equals two to the graph y equals G of X has a slope or that slope is negative 1/8