Question

b) Evaluate the 2^nd order partial derivatives 62 u

6𝑥2

62 u

and 6𝑦2 if

𝑢 = 2𝑥³+3𝑥² 𝑦 + 𝑥𝑦³+𝑦³.

Answer 1

$$u= 2𝑥^3+3𝑥^2 𝑦 + 𝑥𝑦^3+𝑦^3.$$

$$\dfrac{\partial u}{\partial x}=6x^2+6xy+y^3$$

$$\dfrac{\partial^2 u}{\partial x^2}=12x+6y$$

$$\dfrac{\partial u}{\partial y}=3x^2+3xy^2+3y^2$$

$$\dfrac{\partial^2 u}{\partial y^2}=6xy+6y$$

$$\dfrac{\partial^2 u}{\partial x\partial y}=\dfrac{\partial^2 u}{\partial y\partial x}=6x+3y^2$$