[MUSIC].
We've already seen how the chain rule can help us find the anti-derivatives.
Alright, that message is called u substitution.
I'm going to do more messing around with the chain rule and integration.
In particular, let's see what we can do with the chain rule, and the end points
of integration. Well here's an example to play around
with, if I take the integral from 0 to x, of sine of t dt, well that's something.
But let's think about the derivative of this with respect to x.
And by the fundamental theorem of calculus this is sine of x.
The fundamental theorem of calculus is the derivative of the accumulation
function is the integrand. What if that upper endpoint were a
function of x? What I'm asking is what if this endpoint
weren't x anymore? But some function g of x right?
Then this wouldn't be sine of x anymore, be something else.
So to analyze this let's define a function f.
So that function will be given by this rule.
f of x is the integral from 0 to x of sine t dt.
Now what would it mean if we replace that right hand endpoint instead of being x if
we replace that with g of x? That would mean that I want to think
about f of g of x, right? What's f of g of x?
So, that's the integral from 0 to g of x, of sine t dt.
And specifically, right, what I want to know, is what's the derivative with
respect x of f of g of x. That's just the chain-rule.
So let me just write down the chain rule. The derivative of f of g of x is the
derivative of f at g of x times the derivative of g.
So what does it mean in this specific case?
Remember in this case, f of x is the integral from 0 to x of sine t dt.
So the derivative of f, by the fundamental theorem is sin of x and that
means that the derivative of f of g of x, well that's just sin the derivative of f
at g of x times the derivative of g. Let's make it even more concrete.
Let's say that g is the squaring function.
Let me write that down. So if I'm making g the squaring function,
then the derivative of g is just 2x and in that case the derivative of f of g of
x is sine of g which is x squared times the derivative of g which is 2x.
So let's write down our final answer in this specific case.
The final claim is that the derivative of the integral from 0 to g of x which in
this case is x squared of sine t dt. This is the derivative with respect to x
of f of g of x. This is sin of x squared times 2x and
this statement, we can get by combining the chain rule and the fundamental
theorem of calculus.