We're asked to find the partial derivatives
of the given functions.
To find the partial of F with respect to X
we consider Y a constant and differentiate with respect to X
And to find the partial of F with respect to Y,
we consider X a constant and differentiate with respect to Y
Our first function is F[X, Y] equals two
divided by the quantity XY minus four,
raised to the seventh power.
The first step is to rewrite the function
using a negative exponent.
Two divided by the quantity XY minus four
raised to the seventh
is equivalent to two times the quantity XY minus four,
raised to the power of negative seven.
And now to find the partial derivatives,
we will have to apply the chain rule.
We notice how here the inner function,
which we'll label as U,
is the quantity XY minus four.
To find the partial of F with respect to X,
we treat Y as a constant and differentiate with respect to X
So again, we can think of this
as 2U to the power of negative seven.
And then for the derivative with respect to X
is two times negative seven
times U which is really XY minus four,
raised to the power of negative seven minus one,
which is negative eight,
and then times U-prime,
which is a derivative of XY minus four with respect to X
treating Y as constant.
So the derivative of XY with respect to X
treating Y as a constant is just one times Y or Y,
and then minus derivative of four with respect to X
which is zero.
Simplifying,
two times negative seven times Y is negative 14Y.
And let's write this back in fraction form.
So we have divided by the quantity XY minus four
raised to the power of positive eight.
And now to find the partial of F with respect to Y,
we treat X as a constant and differentiate with respect to Y
So, again when you still think of this
as 2U to the power of negative seven.
So, the derivative with respect to Y
would be two times the exponent of negative seven
times U which we know was XY minus four,
raised to the power of negative seven minus one
which is negative eight,
times U-prime.
And now U-prime is the derivative of XY minus four,
with respect to Y, treating X as a constant.
The derivative of XY with respect to Y
treating X as a constant as X times one or just X.
And then minus the derivative of four with respect to Y
which is zero.
Again, simplifying,
two times negative seven times X is negative 14X.
Writing this back into fraction form,
we have divided by the quantity XY minus four,
raised to the power of positive eight.
For the second example we have F[X, Y]
equals E to the power of seven X minus 4Y.
To find the partial derivatives,
we will have to apply the chain rule
where the inner function U is the exponent of 7X - 4Y
Let's write this down on this side.
U is equal to 7X - 4Y.
To find the partial of F with respect to X,
we differentiate with respect to X treating Y as a constant.
The derivative with respect to X
is going to be either U times U-prime,
or U-prime as a derivative of 7X - 4Y
with respect to X treating Y as a constant.
So the partial of F with respect to X
is equal to E to the U or E to the power of 7X - 4Y
and then times the derivative of 7X - 4Y with respect to X
treating Y as a constant.
The derivative of 7X with respect to X is seven.
Because we're treating Y as a constant,
the derivative of negative 4Y with respect to X is zero.
Simplifying,
we have seven times E to the power of 7X - 4Y.
To find the partial of F with respect to Y
we now differentiate with respect to Y
treating X as a constant.
Well, the derivative is still E to the U times U-prime
which is E to the power of 7X - 4Y times U-prime.
But now U-prime
is the derivative of 7X - 4Y with respect to Y
treating X as a constant.
Derivative of 7X with respect to Y is zero
minus the derivative of 4Y with respect to Y
which gives us negative four.
Simplifying,
we have negative four times E to the power of 7X - 4Y
I hope you found this helpful.