# Let f (x, y) = x^3 y + 3 x y^4. Find partial^2 / partial x^2, partial^2 / partial y^2, partial^2...

## Question:

Let {eq}f (x,\ y) = x^3 y + 3 x y^4 {/eq}.

Find {eq}\displaystyle \frac{\partial^2 f}{\partial x^2},\ \frac{\partial^2 f}{\partial y^2},\ \frac{\partial^2 f}{\partial x \partial y},\ \text{and}\ \frac{\partial^2 f}{\partial y \partial x} {/eq}.

## Partial derivative:

Suppose that g is a multivariate function (i.e. having more than one independent variable, x, y, z, etc). The partial derivative of a function g with respect to any of the given independent variable (say x ) is defined as taking the derivative of g as it is a function of x while regarding the other independent variables (here y, z, etc.), as constants.

## Rules for derivatives:

If {eq}g(x){/eq} and {eq}h(x){/eq} are two functions then,

1. The Sum/Difference rule of derivative is {eq}\frac{\partial}{\partial x}\left(g(x,y)\pm h(x,y)\right)=\frac{\partial}{\partial x}\left(g(x,y)\right)\pm \frac{\partial}{\partial x} \left(h(x,y)\right){/eq}.

2. The Product rule of derivative is {eq}\frac{\partial}{\partial x}\left(g(x,y) h(x,y)\right)=\frac{\partial}{\partial x}\left(g(x,y)\right) h(x,y)+g(x,y) \frac{\partial}{\partial x}\left(h(x,y)\right){/eq}.

3. The chain rule of derivative is {eq}\frac{\partial g\left(u(x,y)\right)}{\partial x}=\frac{\partial g}{\partial u} \frac{\partial u(x,y)}{\partial x}{/eq}.