 Welcome to a screencast about the limit definition of the derivative. So the question of today is, how do you compute a derivative of a function using the limit definition? So as a reminder here is our limit definition. So the derivative f prime of x is defined to be the limit as h approaches 0 of f of x plus h minus f of x all over h. So for our example today we're going to be using the function f of x equals x cubed minus 5x plus 4. So f prime of x is going to equal the limit h approaches 0. So our first piece to figure out here is what is f of x plus h? So remember that means we have to take this inside this input x plus h and plug it into our function here and here for x. So when we do that that's going to look like x plus h quantity cubed minus 5 times x plus h plus 4. Then we have a minus sign. Then the second part of our formula is subtracting off the function. So I'm going to put that in parentheses since our function has more than one piece to it. So that's x cubed minus 5x plus 4. And this whole big messiness here is going to be divided by h. Okay, so now comes some algebra skills. Alright, limit h approaches 0. That's going to be constant throughout. X plus h cubed. So thinking back to your algebra days, remember that means x plus h times x plus h times x plus h. Aren't you glad I didn't do a fourth power? Alright, so we have to distribute everything through, combine all of our like terms. So I'll let you try that for make sure you got some algebra practice there. But when you do, you should end up with x cubed plus 3x squared h plus 3h squared x plus h cubed. And again if you have trouble multiplying that out, just make sure like Google it or come and talk to somebody. I'm sure your professor would be happy to help you with that part. Okay, next thing we need to do is distribute our negative 5 through. So that'll be minus 5x minus 5h plus 4. And then last up here on the numerator, we need to distribute our negative sign. So minus x cubed plus 5x and minus 4. And all of this, it's a long division bar, is divided by h. Okay, now comes the fun part. We get to start canceling some terms. So I see an x cubed and a negative x cubed, that was a wipe out. I see a negative 5x and a positive 5x, that was a wipe out. And I see a positive 4 and a negative 4, that was a wipe out. Okay, let's see what we have left. So we have the limit, h approaches 0. We have a 3x squared h plus 3h squared x plus h cubed minus 5h. And this is all divided by h. Okay, so hopefully you notice now in our four terms that are left in the numerator, they all have an h in common. So let's go ahead and factor that out. I prefer to factor it out rather than just start canceling things, because then it's hard to see what's left. So we have 3x squared plus 3hx plus h squared minus 5, once we factor that h out, divided by h. Now we can go ahead and cancel those h's. And we end up with, oops, that's my equal sign up here, limit h approaches 0, 3x squared plus 3hx plus h squared minus 5. So since we now have a nice function, we no longer have our h in the denominator, it's continuous and all that good stuff, if we evaluate this limit, we're going to end up getting 3x squared minus 5. So that is our derivative function when our original function was x cubed minus 5x plus 4. Good luck, thank you for watching.