## Partial differentiation

In the real world, it is difficult to explain behaviour in terms of just one variable.

### Notation

Where the function has two independent variables we can use the following notation: $z=f(x,y)$ $z$ is the dependent variable and $x$ and $y$ are the independent variables.
For example, $z=3x^2 + 2xy - y^2$

Differentiating this function still means the same thing, finding the function for the slope. With more than one variable there is more than one slope.
When we find the partial derivative we differentiate $z$ with respect to $x$ while holding $y$ constant.

The most common notation is: $\frac{\partial z}{\partial x}$ or $\frac{\partial f}{\partial x}$ $(x,y)$

The swirly-d $\partial$ symbol is called “del”.

### Basic rules for partial differentiation

The rules for partial differentiation follow the same logic as univariate differentiation.

Example: $z = 3x^2 + 4xy - y^2$

To find $\frac{\partial z}{\partial x}$, we treat $y$ like a constant and treat $x$ as a variable. $\frac{\partial z}{\partial x}$ $= 6x+4y$

Note: $- y^2$ is a constant as there are no $x$ variables in this term. $4y$ is considered the coefficient of $x$ in the term $4xy$.

To find $\frac{\partial z}{\partial y}$, we treat $x$ like a constant and treat $y$ as a variable $\frac{\partial z}{\partial y}$ $= 4x-2y$

Note: $3x^2$ is treated like a constant when we are differentiating with respect to $y$

## Further information

• Press the Printer Friendly button at the top left-hand corner to download a printable handout
• Khan academy’s introduction to partial derivatives show how to compute the partial derivative and what it means.

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