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# Fractional powers differentiation

## Video transcript

so we have H of X is equal to 5 X to the 1/4 power plus 7 and we want to find what is H prime of 16 or what is the derivative of this function when X is equal to 16 and like always pause this video and see if you can figure it out on your own all right well let's just take the derivative of both sides of this so on the left hand side I'm going to have H prime of X and on the right hand side well the derivative of the right hand side I can just take the derivative of 5 X to the 1/4 and add that to the derivative with respect to X of 7 so the derivative of 5 X to the 1/4 power well I can just apply the power rule here you might say wait wait there's a fractional exponent and I would just say well that's okay the power rule is very powerful so we can multiply the 1/4 times the coefficient so you have 5 times 1/4 X to the 1/4 minus 1 power that's the derivative of 5 X to the 1/4 power and then we have plus 7 now what's the derivative of 7 with respect to X well 7 doesn't change with respect to X the derivative of a constant we've seen this multiple times is just 0 so it's just plus zero and now we just have to simplify this so this is going to be H prime of X is equal to is equal to 5/4 X 2 though what's 1/4 minus 1 well that's negative 3/4 so it's 1/4 minus 4 fourths or negative 3/4 so 5/4 it's X to the negative three-fourths plus 0 so we don't have to write that and now let's see if we can evaluate this when X is equal to 16 so H prime of 16 is 5/4 times 16 to the negative three-fourths well that's the same thing as 5/4 times 1 over 16 to the 3/4 which is the same thing as 5/4 times 1 over let's see I could view this as 16 to the 1/4 and then cubing that and so what is this 16 to the 1/4 is - and then you cube - 2 to the 3rd power is 8 so that's 8 so you have 5/4 times 1/8 which is going to be equal to 5 times 1 is 5 and then 4 times 8 is 4 times 8 is 32 and we are done