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let's say that we have a function f of X and it is equal to negative 4 times the cube root times the cube root of x and what we want to do is we want to evaluate the derivative of our function when X is equal to 8 so see if you can figure this out all right now this might look forward to you might say well I've never taken a derivative of a cube root before but as we'll see we can actually just apply the power rule here because this function can be rewritten as so f of X can be rewritten as negative 4 the cube root of x is the same thing as X to the 1/3 power and now it might be a little bit clearer that we can apply the power rule we could take this 1/3 multiply it by this coefficient negative 4 so we have negative 4 times 1/3 and you have times X to the 1/3 we just decrement that exponent 2 that's a different shade of blue to the 1/3 minus 1 power this is the derivative so f prime of X is equal to that and so now we just have to simplify this is equal to negative 4/3 times X 1/3 minus 1 is negative 2/3 power and so if we want to evaluate F prime of 8 F prime of 8 is equal to negative 4/3 times 8 to the negative 2/3 power well that's the same thing as negative 4/3 times 8 8 to the 1/3 and then raise that to the negative 2 power I'm just using exponent properties here if this looks completely unfamiliar how I got from that to that I encourage you to review exponent properties on Khan Academy well 8 to the 1/3 that is just 2 so this is just 2 and then 2 to the negative 2 power remember let me just take some steps here it's a good review this is equal to negative 4/3 times 2 to the negative 2 is the same thing as one over two to the two these two things are equivalent that and that one over two to the two is the same thing as 2 to the negative 2 so this is 1 over 4 and so this is going to simplify to negative 1 over 3 is equal to negative 1 over 3 and we are done