 Hello and welcome to a video about derivatives of power and constant functions. So the power rule, if you recall, says that if f of x is x to the nth power, then the derivative is f prime of x equals n times x to the n minus 1. So this value of n is just any number, be it a decimal, a fraction, a negative number, anything in there, then this power rule will work. So example one says find the derivative of each of the following functions using proper notation. Okay, so the first function I'm giving you here is g of x equals x to the 2 thirds power. Why don't you go ahead and pause the video for a second, see if you can follow the power rule as it's written up here above and come up with a derivative. So the notation for this one would be g prime of x is going to equal. So we already have this written as a power x to the n, so in this case n is going to be 2 thirds. So I'm going to drop that 2 thirds out front, x to the, then the power rule says you want to subtract 1. So 2 thirds minus 1 is going to give us a negative 1 third. Because remember 2 thirds minus 1 is the same thing as 3 thirds, so 2 thirds minus 3 thirds gives us a negative 1 third. Now there's many different ways you can write this notation then, but that just depends on your instructor and what they expect of you. So I'm just going to leave it as this equation and then you can simplify it from there if necessary. Okay, next equation is p of t equals 1,358. So this is a constant function. I don't see a power in here though, unless you remember the fact that this is really times x to the 0 power. So when you bring that power down, that's going to end up giving us a derivative of 0. Now another way you can think about this too is if you were to graph the function 1,358, it would end up being a horizontal line. Horizontal lines have 0 slope, so of course the derivative is going to be 0. Okay, next one in this example is m of z equals 1 over z to the 2 power or z squared. So for this particular one, we are going to do a little bit of rewriting because the power rule says that our base has to be x and then we have to have a number for our exponent. In this case, I mean zx, variable doesn't matter, but we can't do anything with this 1 over. So we're going to have to go back to our algebra days and rewrite this as z to the negative 2 power. So remember whenever you have a fraction, bring it up, that makes your exponent turn negative. So our derivative in this case, m prime of z, is going to be, so our n is negative 2, so you're going to bring down negative 2, z to the negative 2 minus 1 is going to give us a negative 3. Okay, next example I have, and this just has a little bit different notation. So this d over dx just means take the derivative with respect to x. So in this case, x is our variable. And the function that we want to look at is 1 over the square root of x. So even though the directions look very different than the directions for the first example we did, it's asking the same thing. So find the derivative of this function with respect to x. But again, this one's kind of like the very last one we did. I see a fraction, so we're going to have to do a little bit of rewriting, and I also see a root. Oh goodness, okay, so we're going to have to do a lot of rewriting, but that's okay. So thinking back to your algebra days again, if you have the square root of x, that's defined to be x to the 1 half power. So I can take one step to rewrite this as 1 over x to the 1 half power, and then I can do like we did in the last example. That's going to be x to the negative 1 half power. And you can obviously go from the very first up to the very last up. I'm just trying to break it down a little bit so you can see where things come from. Okay, now I have something that's written as x to a power, so I can go ahead and do my derivative. So in this case, my n is a negative a half. So I have negative a half x to the negative 1 half minus 1. So that's negative 1 half minus 2 halves gives us a negative 3 halves. And that would be our final answer. Again, this can be rewritten just depending on what your instructor expects out of you, but I'm going to leave it like this for now and leave the algebra to you. Thank you for watching.