okay in this video we're going to find derivatives from first principles uh this is one thing to keep in mind when finding the derivative from first principle and that's if you remember this formula here the rest is fairly simple so i'm going to give a few examples and show you how how to incorporate this formula into any given equation okay here we go so our first one is f of x equals four x minus three okay so what we do here anytime you see an x in the function you plug in this here okay so here goes and of course you're evaluating a limit this is part of the the whole process and we use h but as h approaches zero okay so this is how it is now we just plug in this value into um into this so what we have here at is limit as h approaches zero for x plus h minus three so where we saw this x we plug this in and its take away the original function so take away 4x minus 3 so we plug in the x plus h into any x value in the function and it's take away the original function which is here and it's all over h okay so we go ahead and you know work this out so as limit h approaches zero four times x is four x four times h is four h um okay so negative three here of course um keep the negative on the outside and put the original function in in the brackets to make it easier uh to multiply so negative times four x is negative four x negative times negative three x gives us positive three okay and that's all over h so we can go ahead and cross terms out cancel out terms here so negative 4x 4x positive 3 negative 3 so as limit h approaches zero we're left with four h over h okay and right now of course they're dividing you can cancel them out and you're left with four okay in this example we're going to do f of x equals x squared plus 4x okay so as limit h approaches zero okay there's our formula so here we go as limit h approaches zero x plus h squared plus 4 x plus h so anywhere you see an x in the original function of course you plug in x plus h and that's all minus the original function x squared plus four x and that's over h okay and right here we can just expand these brackets so x plus h squared is always x squared plus 2 x h plus h squared and you go ahead and multiply this out so we get 4x plus four h and you multiply this out also negative x squared negative four x and that's over h at this point you can start canceling out terms and dividing okay so um positive 4x negative 4x and positive x squared negative x square okay and we have a few h terms that we can get rid of with this using this here divide this remove that remove one h remove this h also once you get rid of these h's you're left with as limit as h approaches zero 2x plus h plus four okay at this point you apply the h you apply the limit which is zero and any instance of h you encounter you put a zero in so two x plus zero plus four equals two x plus four and that's it