Hi.
Welcome back to recitation.
In lecture you talked about computing derivatives by
definition.
And one rule for computing derivatives that Professor
Jerison mentioned but didn't prove was what's called the
constant multiple rule.
So today I want to give you a proof of that rule and show
you a little bit of geometric intuition for why it works.
So the constant multiple rule says that if you have a
constant c in a differentiable function, f of x, that the
derivative of the function c times f of x is equal to c
times the derivative of f of x.
Just to do a quick example, suppose that c were 3 and f of
x were the function x squared, this says that the derivative
d by dx of 3x squared is equal to 3 times the derivative of d
by dx of x squared.
Now, this is good because we already have a rule for
computing derivatives of powers of x.
So this says we don't need a special rule for computing
multiples of powers of x, where we don't need to go back
to the limit definition to compute the
derivative of 3x squared.
We can just use the fact that we know the derivative of x
squared in order to compute the derivative of 3x squared.
So in this case that would work out to 6x.
In this case.
So it simplifies the number of different computations you
have to do.
It reduces the number of times we need to go back to the
limit definition.
So that's the use of the rule.
Let's quickly talk about its proof.
The idea behind the proof is you have these two derivatives
and you want to show that they're equal.
Well, any time you have a derivative, what it really
means is it's the value of some limit of
some difference quotient.
So in this case we have the derivative d by dx of c times
f of x by definition is the limit of a difference quotient
as delta x goes to 0 of-- so we take the function c times f
of x and we plug in x plus delta x and we plug in x and
we take the difference and we divide by delta x.
So that's c times f of x plus delta x minus c times f of x
divided by delta x.
Now you'll notice that here both terms in the numerator
have this constant factor, c, in them.
So we can factor that out.
And I'll just pull it out in front of this whole fraction
so that this is the limit as delta x goes to 0 of c times
the ratio f of x plus delta x minus f of x, all quantity
over delta x.
Now, c is just some constant.
This part depends on delta x.
And on x, but on delta x.
So as delta x goes to zero, this changes while
this stays the same.
What that means is so as dealt x goes to 0, this gets closer
and closer to something, the value of its limit.
And c--
you're just multiplying it in-- so c times, the limit of
c times this is equal to c times whatever the
limit of this is.
If this is getting closer and closer to some value, c times
it is getting closer and closer to c times that value.
So this is equal to c--
in other words, we can pull constant
multiples outside of limits.
So this limit as delta x, c times the limit is delta x
goes to 0 of f of x plus delta x minus f of x over delta x.
And this limit here is just the definition of the
derivative of f of x.
So this is equal to, by definition, c times d
by dx of f of x.
So we started with the derivative of c times f of x
and we showed this is equal to c times the
derivative of f of x.
That's exactly what we wanted.
So that proves the constant multiple rule.
We've now proved the constant multiple rule--
let me talk a little bit about some geometric intuition for
why this works.
So I've got here, well so, you know, let's take c equals 2,
just for simplicity.
So here I have a graph y equals f of x, and I have also
drawn the graph, y equals 2f of x.
The relationship between these graphs is that y equals 2f of
x is what you get when you stretch the graph for y equals
f of x vertically by a factor of two.
So, you know, if it passed through 0 before, it still
passes through 0.
But everywhere else, if it was above 0, it's
now twice as high.
If it was below 0, it's now twice as low.
So if you think about what the definition, what the
derivative means in terms of this graph geometrically, it's
telling you the limit-- sorry-- the slope
of a tangent line.
Or in other words, the limit of the slopes of secant lines.
So if you look at these two curves, say--
let's pick a couple values of x, say, and then maybe
x plus delta x--
so if you look at the secant lines for these two curves
through those points, what you see is that these two lines,
they have the same, you know-- so the slope of a line is its
rise over its run-- so they have the same run, that we
were talking about the same little interval, here.
But this, in the function that's scaled up in the y
equals 2f of x curve, we have that that the rise--
everything has been stretched upwards by a factor of two--
so the rise here is exactly double the rise here.
So the slope of the secant line is exactly double the
slope of this secant line.
And similarly, the tangent line-- just a limit of secant
lines-- has been stretched by that same factor of two.
So the slope of the tangent line is exactly twice the
slope of the tangent line for the other function.
So the tangent line here is exactly twice as steep as the
tangent line here.
Or in other words, the derivative of this function is
exactly twice the derivative of that function.
So that's just a geometric statement of this very same
constant multiple rule that we stated algebraically at the
beginning and that we just proved.
So that's that.