[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]