U1S4V07 Derivatives of constant multiples.txt

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All right.
We want to think about what happens
when we take a function f of x and we multiply it by 2
to get a new function, which we're calling g of x.
So let's say this is the graph of f of x.
And I've drawn the tangent line here at a point x equals a.
And then to get the graph of g, well,
when we transform f into g all of the y values
are getting doubled.
So we're going to have to stretch the graph of f.
It's going to be stretched vertically by a factor of two.
And when we do that, then the tangent line at this point
is just going to go right along with the stretching.
So it too is going to be stretched vertically.
And when it gets stretched vertically,
that means that the new tangent line to the graph of g
is twice as steep as the tangent line to the graph of f.
So that's telling us that the slope is twice as much.
So g prime of a is twice f prime of a.

This sort of thing is going to hold in general.
So not just if g of x equals twice
f of x, but if it equals any constant times f of x.
So if g of x equals k times f of x for all x,
then the derivative of g is going
to equal k times the derivative of f.
So let's do an example.
We've got g of x equals minus 5 times x squared.
So here we have a constant, minus 5,
being multiplied by a function, x squared.
And we know the derivative of x squared.
So this is good.
We can take the derivative of g just by using this rule.
So g prime of x is going to be-- and then
what we do with the minus 5, since it's multiplying
x squared that's our k.
So the minus 5 is just going to stay in front.
We're going to have minus 5 times the derivative of x
squared.
And I'm going to write that as x squared prime.
And we know the derivative of x squared.
This equals minus 5 times 2x.
And that's just minus 10x.
And that's our answer.
So why don't you try one?