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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.