Chain rule example using visual information | Differential Calculus | Khan Academy

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Differential calculus on Khan Academy: Limit introduction, squeeze theorem, and epsilon-delta definition of limits.

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Closed Caption:


Given capital F of x is equal
to g of x to the third power
where the graph of g and its
tangent line at x equals 4
are shown, what is the
value of F prime of 4?
So they give us g of x
right over here in blue.
And they show us
the tangent line
at x equals 4 right over here.
So we need to figure
out F prime of 4.
So let's just rewrite this
information they've given us.
We know that F of x is equal
to g of x to the third power.
So I'll write it like this,
g of x to the third power.
So we want to figure
out what F prime of x
is when x is equal to 4.
So let's just take
the derivative
here of both sides
with respect to x.
So take the derivative
of the left-hand side
with respect to x, and
take the derivative
of the right-hand side
with respect to x.
So the left-hand
side, this is just
going to be capital
F prime of x.

Now on the right-hand
side I have a composite.
I have g of x to
the third power.
So first we can view this as the
product of the derivative of g
of x to the third power
with respect to g of x.
So there we could
literally just apply
what we know about
the power rule.
The derivative of x to the
third with respect to x is 3x
squared.
So the derivative of gx to the
third with respect to g of x
is just going to be three times
g of x to the second power.
And then we're going to multiply
that times the derivative of g
of x with respect to x.
So times g prime of x
And this comes straight
out of the chain rule.
Derivative of this,
the derivative
of g of x to the third
with respect to g
of x, which is this,
times the derivative
of g of x with respect to x,
which is that right over there.
So now let's just substitute.
We want to figure out
what this derivative is
when x is equal to 4.
So we could say
that F prime of 4
is equal to 3 times g of 4
squared times g prime of 4.
So what is g of 4 going to be?
Well, we can just look at
our function right over here.
When x equals 4, our
function is equal to 3.

So g of 4 is equal to 3.
And what's g prime of 4?
So when x equals
4, g prime of 4 is
the slope of the tangent line.
And they've drawn the tangent
line when x equals 4 here.
So what is the
slope of this line?
So we just have to think about
change in y over change in x.
And I'll look at that between
two integer-valued coordinates.
So it looks like between
these two points.
And when we increase x
by 2, we decrease y by 4.
So as you remember,
slope is rise
over run, or change
in y over change in x.
So the slope of the tangent
line here, the slope
is equal to our
change in y negative 4
over our change in x.
And this is going to
be equal to negative 2.

So this simplifies to F
prime of 4 is equal to-- I'll
do this in a new color--
3 squared is 9 times 3
is 27 times negative 2, which
is equal to negative 54.
So F prime of 4 is negative 54.