We're given f of x, y,
and we're asked to find the following
second order of partial derivatives,
but before we find the second order of partial derivatives,
we need to find the first order of partial derivatives.
So we'll begin by determining the partial derivative
of f with respect to x,
so we'll differentiate f with respect to x,
treating y as a constant.
So the derivative of five x to the sixth
with respect to x would be 30 x to the fifth
plus the derivative of four xy to the third
with respect to x would just be four y to the third.
Again, the derivative of four x would be four,
and we're treating y to the third as a constant,
and then plus the derivative of six y to the second
with respect to x would be zero,
again, because we're treating y as a constant.
And then we'll find the first partial derivative
of f with respect to y, so we'll differentiate
with respect to y, treating x as a constant.
So the derivative of five x to the sixth
with respect to y would be zero.
The derivative of four x y to the third
with respect to y, we'd multiply by three,
subtract one from the exponent.
That would be 12 xy squared, and then the derivative
of six y squared with respect to y would be 12y.
And now we want to find the second order of
partial derivative with respect to x,
which means we want to differentiate
f with respect to x twice, treating y
as a constant both times, which means
now we'll find the partial derivative
of this partial derivative with respect to x again.
So the derivative of 30 x to the fifth
with respect to x would be 150 x to the fourth,
and the derivative of four y to the third
with respect to x would be zero.
Next, we have the second order of partial derivative
with respect to y, so we want to
differentiate f with respect to y twice,
treating x as a constant both times,
which means now we'll differentiate
this partial derivative with respect to y again.
So the derivative of 12x y to the second
with respect to y, we would multiply it by two
and subtract one from the exponent, giving us 24xy.
Then the derivative of 12y
with respect to y would be just 12.
And now these next two partial derivatives
are called mixed partials.
For this first mixed partial, we want to differentiate
f with respect to x first, and then y,
which means we'll find the derivative
of this partial derivative with respect to y,
so the derivative of 30 x to the fifth
with respect to y would be zero,
and then the derivative of four y to the third
with respect to y would be 12y squared.
For the next mixed partial derivative, we would
differentiate f with respect to y first,
and then x, which means we'll find the partial derivative
of this partial derivative, now with respect to x.
So the derivative of 12x y to the second with respect to x
would be 12y squared, and the derivative of 12y
with respect to x would be zero.
And notice how these two mixed partial derivatives
are equal, and this will be true as long as f in the
first partial and second partial derivatives are continuous.
Before we go, let's talk about the meaning
of these second order partial derivatives.
The sign of the second order partial derivative
with respect to x indicates the concavity
of f of x,y in the x direction.
If it's positive, it's concave up in the x direction,
and if it's negative, it's concave down the x direction.
And the sign of the second partial with respect to y
indicates the concavity in the y direction.
So again, if the sign of this is positive,
f would be concave up in the y direction,
and if it's negative, it would be
concave down in the y direction.
And the mixed partial derivatives tell us how
a partial derivative in one variable is changing
in the direction of the other.
We can also say that the mixed partial derivative
with respect to x and then y tells us
how the rate of change of f in the x direction
is changing as we move in the y direction.
I hope you found this helpful.