Question

Obtain all the first and second order partial derivatives of the function:
f( x, y) =ln( x^2 + y^2)

Answer 1

$$f( x, y) =ln( x^2 + y^2)$$

$$\dfrac{\partial f} {\partial x}( x, y) =\dfrac{2x}{ x^2 + y^2}$$

$$\dfrac{\partial f} {\partial y}( x, y) =\dfrac{2y}{ x^2 + y^2}$$

$$\dfrac{\partial^2 f} {\partial x^2}( x, y) =2\dfrac{y^2-x^2}{ (x^2 + y^2)^2}$$

$$\dfrac{\partial^2 f} {\partial y \partial x}( x, y) =\dfrac{-4xy}{ (x^2 + y^2)^2}$$

$$\dfrac{\partial^2 f} {\partial x \partial y}( x, y) =\dfrac{-4xy}{ (x^2 + y^2)^2}$$

$$\dfrac{\partial^2 f} {\partial y^2}( x, y) =2\dfrac{x^2-y^2}{ (x^2 + y^2)^2}$$