Question

Find the first partial derivatives of the function.
f(r,s) = r ln(r^2+s^2).

Answer 1

There are two first partial derivatives of
`f(r,s) = r\cdot \ln (r^(2)+s^(2))`:
`\frac{\partial{f}}{\partial {r}} = \ln(r^(2)+s^(2)) + (2\cdot r^(2))/(r^(2)+s^(2))` and
`\frac{\partial {f}}{\partial{s}} = (2\cdot r\cdot s)/(r^(2)+s^(2))`.

Explanation:

Let be
`f(r,s) = r\cdot \ln (r^(2)+s^(2))`. The quantity of first partial derivatives of a multivariate function is equal to the number of variables. For this function, there are two first partial derivatives:

`\frac{\partial{f}}{\partial {r}} = \ln(r^(2)+s^(2)) + (2\cdot r^(2))/(r^(2)+s^(2))`

`\frac{\partial {f}}{\partial{s}} = (2\cdot r\cdot s)/(r^(2)+s^(2))`