We were given z of x, y,
where x equals at 9s minus 7t,
and y equals negative 2s plus 5t.
We're asked to calculate the partial derivative
of z with respect to s,
and the partial derivative of z with respect to t
by first finding the partial of x with respect to s,
the partial of y with respect to s,
the partial of x with respect to t,
and the partial of y with respect to t.
We will need all of these
because we'll have to apply the chain rule shown below
in order to determine the partial of z with respect to s
and the partial z with respect to t.
So let's begin by determining the partial of x
with respect to s,
and the partial of x with respect to t.
To find the partial of x with respect to s,
we differentiate x with respect to s,
treating t as a constant,
and therefore the partial of x with respect to s is nine.
To find the partial of x with respect to t,
we differentiate x with respect to t,
now treating s as a constant.
The partial of x with respect to t is negative seven.
Let's also find the partial of y with respect to s,
and the partial of y with respect to t.
To find the partial of y with respect to s,
we differentiate y with respect to s,
treating t as a constant,
which gives us negative two.
And to find the partial of y with respect to t,
we differentiate y with respect to t,
treating s as a constant, which gives us five.
And now let's begin determining
the partial of z with respect to s.
This is equal to the partial of z with respect to x
times the partial of x with respect to s,
plus the partial of z with respect to y,
times the partial of y with respect to s.
We begin with the partial of z with respect to x.
So we differentiate z with respect to x,
treating y as a constant, which gives us negative 6x.
And then we have times the partial of x with respect to s,
which we already know is positive nine.
And then we have plus the partial of z with respect to y.
So now differentiating z with respect to y,
treating x as a constant, we have 18y,
times the partial of y with respect to s,
which we already found is negative two.
Now we can't leave the partial derivative in this form.
We want to express it in terms of s and t, not x and y.
So now we substitute 9s minus 7t for x,
and negative 2s plus 5t for y.
When doing this, notice
negative six times nine is negative 54.
So we have negative 54 times x,
which is 9s minus 7t.
18 times negative two is negative 36.
Let's write this as minus 36 times y,
where y is negative 2s plus 5t.
So while we could leave the derivative in this form,
let's go ahead and put the parentheses
and combine like terms.
Distributing negative 54,
negative 54 times 9s is negative 486s.
Negative 54 times negative 7t is equal to positive 378t.
And then because of the subtraction,
let's distribute negative 36.
Negative 36 times negative 2s gives plus 72s,
and negative 36 times positive 5t gives us minus 180t.
For the last step, we combine like terms.
Negative 486s plus 72s
is negative 414s.
And 378t minus 180t
equals 198t, giving plus 198t.
Let's go and record this.
And now let's determine the partial of z with respect to t.
The partial of z with respect to t is equal
to the partial of z with respect to x
times the partial of x with respect to t,
plus the partial of z with respect to y
times the partial of y with respect to t.
So we begin with the partial of z with respect to x,
which we found earlier.
Differentiating z with respect to x,
treating y as a constant, we have negative 6x.
And then times the partial of x with respect to t,
which is negative seven,
plus the partial of z with respect to y,
which again, we found earlier.
Differentiating z with respect to y,
treating x as a constant, gives us 18y,
and then times the partial of y with respect to t,
which we already found is five.
And again, we perform substitution for x and y.
Negative six times negative seven is 42,
and x, again, is 9s minus 7t.
18 times five is 90,
giving us plus 90 times y,
and y is equal to negative 2s plus 5t.
Again, let's go ahead and clear the parentheses
and combine like terms.
42 times 9s is 378s.
42 times negative 7t gives us minus 294t.
90 times negative 2s gives us minus 180s,
and that 90 times 5t gives us plus 450t.
And last step, combining like terms,
378s minus 180s equals 198s,
and a negative 294t plus 450t
is equal to 156t, giving us plus 156t.
And let's go and record this as well.
I hope you found this helpful.