[MUSIC] Here is the question that I want to address right now.
What's the derivative of some constant multiple of some function? Now, in this
case the constant multiple is 2 but of course that 2 could be replaced by any
sixth number. If you don't like the D D X notation,
another way to ask this question is this. If you've got some new function G and
it's the constant multiple times F, again in this case I'm using 2 as the constant,
the question is, what's the derivative of G in terms of the derivative of F?
To gain some intuition, let's pick a specific example, and look at a graph.
So here's the graph, of, just some random function.
Let's suppose that I stretch this graph, in the y direction.
So now I stretch the y axis, and that corresponds to multiplying the function
by a constant value. In this case, to,
how do the tangent lines change when I do this stretching? So if I double the Y
axis, the function changes by twice as much for the same input change.
So if I double the Y axis, the slope of the tangent line also doubles and that
makes sense numerically. Here's what I know.
I know that g of x is twice f of x. G is this constant multiple of f.
I also know something about the derivative of f.
The derivative encodes how input changes become output changes, or, a bit more
precisely, the derivative in the limit is the ratio of output change to input
change. So if I multiply the ratio of output
change to input change by an actual input change, this at least approximately is
telling me how much the output should change when I move from x to x plus h.
Right, f's new output at the input x plus h.
Is it's old output plus how much I expect the output to change.
This is a really nice way to summarize what the derivative's saying.
I know another thing. I know that g of x plus h is twice f of x
plus h just because g is twice f for any input value x, so in particular that's
true when the input is x plus h. These two statements are connected.
Alright, G of X plus H is twice F of X plus H and F of X plus H is approximately
this. So I can combine those two statements
together in this statement. G of X plus H is about twice f(x)+h is
approximate value, right. 2f(x)+2h*f-prime(x), which I've written
as h*2f-prime(x). I made this a little bit nicer.
Since 2f(x) is g(x), I can replace this 2*f(x), with g(x).
And this is really looking good. This is telling me that g's output at x+h
is about, g's output at x, plus how much I change the input by, times some
quantity. Now, considering that the actual
derivative of g would tell me some information like this.
That g's output is about, g's old output plus how much I change the input by times
the derivative. You're beginning to see what's going on
here, right? Look, I've got the derivative of g here.
And I've got twice the derivative of f here.
And if you sort of believe that these statements are connected in this way, you
might then believe that the derivative of g is twice the derivative of f.
The derivative of g here is twice the derivative of f.
We can formalize this as a rule. Here's the constant multiple rule.
So let k be constant and suppose that f is just some function which is
differentiable at the point a. G is that constant multiple of f, so g of
x is k*f(x). Given this setup, what the constant
multiple rule is concluding, is that the derivatives are related in the same way.
The derivative at the point A is K times the derivative of F at the point A.
Now if you don't like this prime notation you could also write it using the D D X
notation. So here's how I write the constant
multiple law using the D D X notation. The derivative of k times a function is k
times the derivative of that function. I encourage you to keep practicing.
With time, you'll be able to calculate the derivative of just a ton of different
functions. [MUSIC]