If z=1/(x^2+y^2-1)
is x(dz/dx)+y(dz/dy)=-2z(1+z) or -z^2(1+z)
note:dz/dx=partial derivative of x
and dz/dy=partial derivative of y??
∂z/∂x = -2x/(x^2+y^2-1)^2
∂z/∂y = -2y/(x^2+y^2-1)^2
so,
x ∂z/dx + y ∂z/dy = -2(x^2+y^2)/(x^2+y^2-1)^2
= -2z^2(1+z)
To determine whether x(dz/dx) + y(dz/dy) is equal to -2z(1+z) or -z^2(1+z), we need to find the partial derivatives dz/dx and dz/dy first.
Given z = 1/(x^2 + y^2 - 1), we can find the partial derivatives as follows:
Partial derivative dz/dx:
To find dz/dx, we differentiate z with respect to x while treating y as a constant.
Using the quotient rule, we have:
dz/dx = (-1)(2x)/((x^2 + y^2 - 1)^2)
= -2x/(x^2 + y^2 - 1)^2
Partial derivative dz/dy:
To find dz/dy, we differentiate z with respect to y while treating x as a constant.
Using the quotient rule again, we have:
dz/dy = (-1)(2y)/((x^2 + y^2 - 1)^2)
= -2y/(x^2 + y^2 - 1)^2
Now, let's substitute these partial derivatives into x(dz/dx) + y(dz/dy) and simplify it to compare it with -2z(1+z) and -z^2(1+z):
x(dz/dx) + y(dz/dy) = x(-2x/(x^2 + y^2 - 1)^2) + y(-2y/(x^2 + y^2 - 1)^2)
= -2x^2/(x^2 + y^2 - 1)^2 - 2y^2/(x^2 + y^2 - 1)^2
= -2(x^2 + y^2)/(x^2 + y^2 - 1)^2
Comparing this with -2z(1+z), we have:
-2z(1+z) = -2(1/(x^2 + y^2 - 1))(1 + 1/(x^2 + y^2 - 1))
= -2(1/(x^2 + y^2 - 1))^2 - 2/(x^2 + y^2 - 1)
Therefore, x(dz/dx) + y(dz/dy) is equal to -2z(1+z), not -z^2(1+z).
To determine whether x(dz/dx) + y(dz/dy) equals -2z(1+z) or -z^2(1+z), we need to calculate the partial derivatives dz/dx and dz/dy first.
Given z = 1/(x^2 + y^2 - 1), let's find the partial derivative dz/dx:
1. Keep y constant and differentiate z with respect to x.
dz/dx = - (2x) / (x^2 + y^2 - 1)^2
Next, let's find the partial derivative dz/dy:
1. Keep x constant and differentiate z with respect to y.
dz/dy = - (2y) / (x^2 + y^2 - 1)^2
Now we can substitute the partial derivatives back into the equation x(dz/dx) + y(dz/dy).
x(dz/dx) + y(dz/dy) = x * (-(2x) / (x^2 + y^2 - 1)^2) + y * (-(2y) / (x^2 + y^2 - 1)^2)
Factor out -2 from both terms:
= -2 * (x^2 / (x^2 + y^2 - 1)^2) - 2 * (y^2 / (x^2 + y^2 - 1)^2)
= -2 * ((x^2 + y^2) / (x^2 + y^2 - 1)^2)
Now, let's simplify the numerator of the equation.
x^2 + y^2 = (x^2 + y^2 - 1) + 1
= (x^2 + y^2 - 1) + 1
= (x^2 + y^2 - 1) + (x^2 + y^2 - 1 + 2)
= (x^2 + y^2 - 1) + (x^2 + y^2 + 1)
= 2(x^2 + y^2)
Substituting this back into the equation:
= -2 * (2(x^2 + y^2) / (x^2 + y^2 - 1)^2)
= -4(x^2 + y^2) / (x^2 + y^2 - 1)^2
Therefore, x(dz/dx) + y(dz/dy) = -4(x^2 + y^2) / (x^2 + y^2 - 1)^2.
So, the original equation x(dz/dx) + y(dz/dy) equals -4(x^2 + y^2) / (x^2 + y^2 - 1)^2.