We're asked to find the partial derivatives
of the given functions.
The partial derivative with respect to x is a function
that gives the slope in the x-direction
and the partial derivative with respect to y is a function
that gives the slope in the y-direction.
To determine the partial derivative of f with respect to x,
we consider y a constant
and differentiate with respect to x.
To determine the partial derivative of f with respect to y,
we consider x a constant
and differentiate with respect to y.
So for the first example,
we're given f of x comma y equals 1,200
plus seven x minus three y.
To find the partial of f with respect to x,
we treat y as a constant
and differentiate with respect to x.
To begin, the derivative of 1,200 respect to x
is equal to zero.
Remember the derivative of any constant is equal to zero
and then we have plus the derivative of seven x
with respect to x, which is seven.
And then we have minus a derivative of three y
with respect to x, because we are treating y as a constant,
three y is a constant, and therefore the derivative
with respect to x is zero.
The partial of f with respect to x is equal to seven.
Next we want to find the partial of f with respect to y.
So we treat x as a constant
and differentiate with respect to y.
The derivative of 1,200 with respect to y is zero
plus the derivative of seven x
with respect to y is also zero.
Again because we treat x as a constant
and therefore seven x is also a constant.
And then we have minus the derivative of three y
with respect to y which gives us negative three.
The partial of f with respect to y is negative three.
Next we have f of x comma y equals four x to the eighth
minus two y to the sixth plus x plus y.
We begin by determining the partial of f with respect to x,
so we treat y as a constant
and differentiate with respect to x.
The derivative of four x to the eighth with respect to x
is four times eight x to the seventh,
and then we have minus the derivative of two y to the six
with respect to x.
Because we treat y as a constant,
the derivative with respect to x is zero
plus the derivative of x with respect to x which is one
giving us plus one and then the derivative of of y
with respect to x is equal to zero,
again because we are treating y as a constant
and differentiating with respect to x.
And therefore the partial of f with respect to x
is equal to 32x to the seventh plus one.
Next we want to find the partial of f with respect to y,
so now we treat x as a constant
and differentiate with respect to y.
The derivative of four x to the eighth
with respect to y is equal to zero,
again because we're treating x as a constant.
And then we have minus the derivative of of two y to the six
with respect to y which is equal to negative two
times six y to the fifth plus the derivative of x
with respect to y.
Because we treat x as a constant,
the derivative with respect to y is zero
and then plus the derivative of y with respect to y
which is one giving us plus one,
simplifying we have negative 12y to the fifth
plus one for the partial with respect to y.
For the last example we have f of x comma y
is equal to nine x to the fifth, y to the ninth.
To find the partial of f with respect to x,
we differentiate with respect to x treating y as a constant.
And therefore the derivative of nine x to the fifth,
y to the ninth with respect to x
is nine times five x to the fourth
and we still have times y to the ninth
because we are treating y as a constant.
And therefore the final partial with respect to x
is 45x to the fourth, y to the ninth.
And now to find the partial of f with respect to y,
we treat x as a constant
and differentiate with respect to y.
The derivative of nine x to the fifth, y to the ninth
with respect to y is nine x to the fifth
times nine y to the eighth
which gives us 81x to the fifth, y to the eight
for the partial of f with respect to y.
I hope you found this helpful.