We're given the function f of x comma y,
and asked to find the partial derivatives
and evaluate at the given point.
Notice how we have two first order partial derivatives
and two second order partial derivatives.
Let's begin by reviewing
what the first order partial derivatives tell us
about our surface, as well as how to find them.
The first order partial derivative
of a function of two variables
tells us the slope of the tangent line at a given point
in the either the positive x or positive y direction.
This measures the rate of change of z,
or a function f with respect to x,
or the rate of change of z, or f, with respect to y,
based upon which partial derivative we're determining.
To determine the partial derivative with respect to x,
we consider y a constant
and differentiate with respect to x.
To determine the partial derivative with respect to y,
we consider x a constant
and differentiate with respect to y.
So, going back to our example,
we'll begin by determining a partial derivative
with respect to x.
So, we'll differentiate each term with respect to x,
treating y as a constant.
So, the derivative of five x squared with respect to x
would be 10x,
plus the derivative of three x squared
y cubed with respect to x,
treating y cubed as a constant.
So, we'd multiply by two,
and then subtract one from the exponent on x,
which would give us six x to the first,
or six x, y to the third.
And then the derivative of three y to the fifth
with respect to x would be zero.
Again, because you're treating y as a constant.
So, here is our first order partial derivative
with respect to x, and now we'll evaluate this at our point.
Notice how we only need the x and y values.
So, we'll substitute 1/2 for x, and one for y.
So, we'd have 10 times 1/2,
plus six times 1/2,
times one cubed,
which would give us five,
plus three,
which equals eight,
which means the slope of the tangent line in the x direction
at this point is positive eight.
And let's look at this graphically.
Our surface is graphed here in purple.
The point we're considering is this red point,
and this blue line is tangent line at that point
in the x direction, which is in this yellow plane,
which is parallel to the x-axis.
This arrow here indicates the positive x direction.
Notice, as we move in this direction from our point,
the line is moving uphill,
and it has a slope of positive eight.
So, this is what the partial derivative with respect to x
is telling us about the tangent line at that point.
Now, we want to find the
partial derivative with respect to y.
So, we'll differentiate with respect to y,
treating x as a constant.
So, the derivative of five x squared with respect to y
would be zero.
The derivative of three x squared y cubed with respect to y,
we would multiple by three,
so that would give us nine.
Then we'd have x squared y to the second
because we subtract one from the exponent of three on the y.
And then plus the derivative of three y to the fifth
with respect to y would be 15 y to the fourth.
And now, we'll evaluate this at our point.
So again, we'll substitute 1/2 for x, and one for y.
So, we'll have nine times 1/2 squared,
times one squared,
plus 15 times one to fourth.
So here, we'd have 9/4
plus 15,
15 over one,
common denominator would be four.
So, we have 9/4 plus 60/4, or 69/4.
So, the slope of the tangent line
in the positive y direction at this point
would be positive 69/4.
Let's take a look.
Again, here is a graph of our surface.
Now the plane is parallel to the y-axis,
and this arrow here indicates
the direction of the positive y-axis.
So, notice how the tangent line is graphed here in blue.
Notice if we move in the positive y direction,
the slope of the tangent line would be positive 69/4.
Notice how it's going uphill very quickly.
Now, let's review what the second order partial derivatives
tell us about a surface and how to find them.
If the second order partial with respect to x is positive,
then our function is concave up in the x direction.
If the second order partial with respect to y is positive,
then the function is concave up in the y direction.
And of course, if these are negative,
this would indicate that the function would be concave down
in the x and y directions.
The mixed partials, indicated using this notation here,
tells us how a partial in one variable
is changing in the direction of the other.
So, the partial derivative of f with respect to x,
and then with respect to y,
tells us how fast the rate of change of f in the x direction
is changing as we move in the y direction.
We could also say this tells us
how the slope of the tangent line
in the x direction is changing
as we move in the y direction.
And the mixed partials are equal
if both mixed partials are continuous
on an open region for every point in that region.
So, to find the second order partial with respect to x,
we want to differentiate with respect to x twice.
So, we'll take our first order partial
with respect to x here,
and differentiate with respect to x again,
treating y as a constant.
So, the derivative of 10x with respect to x would be 10,
plus the derivative of six x,
y to the third with respect to x,
so we're treating y to the third as a constant.
So, our derivative with respect to x
would just be six y to the third.
And now, we'll evaluate this at our point.
So, we'd have 10 plus
six times one to the third,
which equals 16.
So, notice how the second order partial with respect to x
is positive, or greater than zero,
meaning from this point, if we move in the x direction,
the function is concave up.
And let's go ahead and verify that.
Notice from this view,
where the yellow plane is parallel to the x-axis,
we can see that the surface is concave up
in the x direction.
And then finally, we want to find
the partial derivative with respect to x,
then with respect to y, which means now we'll differentiate
this first order partial with respect to x
with respect to y.
So, we'll treat x as a constant.
So, the derivative of 10x with respect to y would be zero.
The derivative of six x, y to the third
with respect to y would be 18, x, y squared.
So, let's evaluate this at our point.
So, we'd have 18 times 1/2,
times one squared.
So, this would be nine.
And again, notice how this is
positive, or greater than zero,
which means the slope of the tangent line
in the x direction, as we move in the y direction,
is increasing.
And before we go, let's go ahead and find
the other second order mixed partial.
The partial derivative of f with respect to y,
then with respect to x,
which means we would now find the derivative
of this partial derivative with respect to x.
So, we treat y as a constant.
So, the derivative of nine x squared, y squared
with respect to x would be 18, x to the first,
y squared, or 18,
x, y squared.
And the derivative of 15, y to the fourth with respect to x
would be zero.
So, notice how our second order and mixed partials
are equal.
I hope you found this helpful.