, In our quest for a function which with it's own derivative, we met e to the x.
Remember, the derivative of e to the x is e to the x. What's the inverse function
for e to the x? What function undoes that sort of exponentiation? Well, we really
don't have a name for that function yet, so we're just going to call it Log.
So in symbols, if e to the x is equal to y, then log y equals x, right?
Log is the inverse function for e to the, the log of something I must raise e to get
back the thing I plugged into log. These logs or logarithms are super important for
a ton of reasons. Take a look at this. Since e to the x plus y is e to the x
times e to y, right? This is the property of exponents, if you like. There's a
corresponding statement about log. Log of a times b is log of a plus log of b. Or, a
shorthand way to say that is that logarithms transform products into sums.
This is a big reason why we care so much about logs. Once we've got this new
function, log, we can ask what's the derivative of log? So, if f of x is e to
the x, the inverse function is log. If I want to now differentiate log, I can
use the inverse function theorem. So, the derivative of the inverse function is 1
over the derivative of the original function evaluated at the inverse function
of x. Now, the neat thing here is that the derivative of e to the x is itself. So, f
prime is just f, and I'm left with f of f inverse of x. It's e to the log of x. But
log of x tells me what I have to plug into e to get out the input, right? f of the
inverse function of f is just, it would be the, the same input again. So, this is 1
over x. So, the derivative of log x is just 1 over x. And you can really see this
fact on the graph. Here's a graph of y equals log x. And I should warn you right
off the bat that the x-axis and the y-axis have totally different scales. The x-axis
goes from 1 to 100. The y-axis in this plot goes from 0 to 5. it's going to make
it not so easy to tell the exact values of the slopes and tangent lines, but you can
see from this graph the important qualitative feature. That the graph is
getting less and less slopey. And if you like, it's flattening out as the input
gets bigger. if I put down a tangent line and I start moving the point that I'm
taking the tangent line at to the right, you can see the tangent line slope is
getting closer and closer to zero. And, of course, that's reflected by knowing the
derivative of log x is 1 over x. So, if x is really big, the tangent line at x is
really close to zero in slope. Think about log of a really big number. For instance,
what's log of a million? A log of a million is about 13.815510. And, of
course, it keeps going. I, it's an irrational number. But, now the derivative
of log, right? Is 1 over its input. So, what does that tell you that you might
think log of a million and 1 is equal to? Well, the derivative tells you how much
wiggling input affects the output. So, if I wiggle the input by 1, you expect the
output to change by about the derivative. And yeah, log of a million and 1 is about
13.815511, right? What's being affected here is in the millionths place after the
decimal point, right? It's the 6th digit after the decimal point because it's being
affected like a change of 1 over a million. All right? I'm changing the
output by about a millionth. At this point, we can also handle logs with other
bases. So, let's suppose I want to differentiate log of x base b, right? This
is the number that I'd raise b to, to get back x. Well, there's a change of base
formula for log. This is the same as the derivative of, say, the natural log of x
over the log of b. But the log of b is a constant, and the derivative of a constant
multiple is just that constant multiple times the derivative. So, this is 1 over
log b times the derivative of, here's a natural log of x. But I know the
derivative of the natural log of x, it's 1 over x.
So, the derivative of log of x base b is 1 over log b times 1 over x. Or maybe
another way to write this would be 1 over x times log b, if you prefer writing it
that way. e to the x is a sort of key that unlocks how to understand the derivative
of a ton of other exponential functions. For example, now that we know how to
differentiate e to the x, we can also differentiate 2 to the x. So, let's
suppose I want to differentiate 2 to the x. Now, you might just memorize some
formula for differentiating this. But it's easier, I think better, to just recreate
this function out of the functions that you already know all the derivatives of.
So, in this case, let's replace 2 by e to the log 2 to the x, right? So, instead of
writing 2 here, I've just written e to the log 2, this is just 2. But I've got e to
the log 2 to the x and that's the same as e to the log 2 times x. You know, this is
a composition of functions that I know how to differentiate. I know how to
differentiate e to the, and I know how to differentiate constant multiple times x.
So, by the chain rule, it's the derivative of the outside function. So, which is
itself, e to the, at the inside function, which is log 2 times x, times the
derivative of the inside function which in this case is log 2 log x. So, I'm just
going to multiply by log 2. Now, I could kind of make this look a little bit nicer,
right? e to the log 2 times x, well, that's just 2 to the x times, again log 2.
So, the derivative of 2 to the x is 2 to the x times log 2. And, of course, 2
didn't play any significant role here. I could have replaced 2 by any other number
and I'd get the same kind of formula. What I hope you're seeing is that all of the
derivative laws are connected. With practice, you'll be able to differentiate
any function that you build by combining our standard library of functions and
operations on those functions.