z = 9u / u2 + 5v find dz/du and dz/dv

To find the partial derivatives dz/du and dz/dv, we first need to understand the function z. The function z is given as z = 9u / (u^2 + 5v).

1. To find dz/du, we need to apply the quotient rule which is (v * du/dx - u * dv/dx) / v^2. Here, u = 9u and v = u^2 + 5v.

   So, dz/du = [(u^2 + 5v) * d(9u)/du - 9u * d(u^2 + 5v)/du] / (u^2 + 5v)^2 = [(u^2 + 5v) * 9 - 9u * 2u] / (u^2 + 5v)^2 = [9u^2 + 45v - 18u^2] / (u^2 + 5v)^2 = [-9u^2 + 45v] / (u^2 + 5v)^2

2. To find dz/dv, we again apply the quotient rule.

   So, dz/dv = [(u^2 + 5v) * d(9u)/dv - 9u * d(u^2 + 5v)/dv] / (u^2 + 5v)^2 = [0 - 9u * 5] / (u^2 + 5v)^2 = -45u / (u^2 + 5v)^2

So, the partial derivatives dz/du and dz/dv are [-9u^2 + 45v] / (u^2 + 5v)^2 and -45u / (u^2 + 5v)^2 respectively.