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# Derivatives expressed as limits

Video transcript

- [Voiceover] Let's see
if we can find the limit as h approaches zero of
five log of two plus h minus five log of two, all of that over h. And I'll give you a little bit of a hint because I know you're
about to pause the video and try to work through it. Think of your derivative properties, especially the derivatives
of logarithmic functions, especially logarithmic
functions in this case with a base 10. If someone just writes
log without the base, you can just assume that that
is a 10 right over there. So pause the video, and see
if you can work through it. Alright, so the key here
is to remember that if I have, if I have f of x, let me do it over here, I'll do it over here. F of x and I want to find f prime of, let's say f prime of some number, let's say a, this is going to be equal to the limit as x, or sorry, as h approaches zero is one
way of thinking about it, as h approaches zero of f of a plus h, minus f of a, all of that over h. So this looks pretty close
to that limit definition, except we have these fives here. But lucky for us we can
factor out those fives, and we could factor them out, we could factor them out out front here, but if you just have a scale
or times the expression, we know from our limit properties we can actually take those
out of the limit themselves. So let's do that, let's take both of these fives, and factor them out, and so this whole thing
is going to simplify to five times the limit as h approaches zero of log of two plus h, minus, minus log of two, all of that over h. Now, you might recognize
what we have in yellow here, cause think about it, what this is, if we had f of x is equal to log of x, and we wanted to know what f prime of, well actually let's say f prime of two is, well this would be the limit as h approaches zero of log of two plus h, two plus h, minus log of two, minus log of two, all of that over h. So this is really just a, what we see there, this by definition this right over here is, is f prime of two. If f of x is log of x, this is f prime of two, f prime of two. So can we figure that out? If f of x is log of x, what is f prime of x? F prime of x, we don't need
to use the limit definition. In fact the limit definition is quite hard to evaluate this limit, but we know how to take the derivative of logarithmic functions. So f prime of x is going to be equal to one over the natural log of our base, our base here we already
talked about that, that is 10. So one over natural log of 10, times, times, times x. If this was a natural log, well, then, this would be one over natural log of e times x, natural log of e is just one, so that's where you get the one over x, but if you have any other base you put the natural log of that base right over here in the denominator. So what is f prime of two? F prime of two is one over the natural log of 10, times two. So this whole thing has simplified, this whole thing is equal
to five times this business. So I could actually just write it as it's equal to five over, five over the natural log of 10, natural log of 10, times two, I could have written it
as two natural log of 10s. The key here for this type of exercise, you might be a little let me see if I can evaluate this limit? You're like whoa, this looks
a lot like the derivative of a logarithmic function, especially the derivative
when x is equal to two, if we could just factor these fives out. So you factor out the fives, and say hey, this is,
this is the derivative of log of x when x is equal to two, and so we know how to take
the derivative of log of x. If you don't know we have
videos where we prove this. Where you take the
derivatives of logarithms with bases other than e, and you just use that to actually and you find the derivative, and you evaluate it at two, and then you're done!