Derivative of a multi-variable function: the Partial Derivative

Let's say we have a function:

$$f(x_1,x_2,...,x_i, ...,x_n)$$

Then we can take the derivative for each variable:

$$\begin{aligned} \frac{\partial f}{\partial x_1} &= \lim_{h\to0}\frac{f(x_1+h,x_2,...,x_i, ...,x_n) - f(x_1,x_2,...,x_i, ...,x_n)}{h}\\ \frac{\partial f}{\partial x_2} &= \lim_{h\to0}\frac{f(x_1,x_2+h,...,x_i, ...,x_n) - f(x_1,x_2,...,x_i, ...,x_n)}{h}\\ ...\\ \frac{\partial f}{\partial x_i} &= \lim_{h\to0}\frac{f(x_1,x_2,...,x_i+h, ...,x_n) - f(x_1,x_2,...,x_i, ...,x_n)}{h}\\ ...\\ \frac{\partial f}{\partial x_n} &= \lim_{h\to0}\frac{f(x_1,x_2,...,x_i, ...,x_n+h) - f(x_1,x_2,...,x_i, ...,x_n)}{h} \end{aligned}$$

In words:

The partial derivative is the derivative of a multi-variable function taken by keeping but one of its variables constant. Thus the partial derivative of $f(x_1,x_2,...,x_i, ...,x_n)$ to $x_i$, thus $\frac{\partial f}{\partial x_i}$, is the derivative of the function with $x_i$ left variable and $x_1$, $x_2$, ..., $x_n$ kept constant.

A sample

Let us define the following function (we're using $x$ and $y$ instead of $x_1$ and $x_2$ to avoid subscript hell):

$$f(x, y) = 0.005(x^3 + y^3)$$

Then we can write the partial derivatives as:

$${}$$

You can use the following slider to change the coordinate $x$ at which the derivative is taken

You can use the following slider to change the coordinate $y$ at which the derivative is taken

In following, the

Purple lines are the points of constant $x$ and $y$. (These are the values you influence with the sliders)
Red lines are the functions at a respective constant $x$ and $y$
Green lines are the respective partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$

The function $$f(x, y) = 0.005(x^3 + y^3)$$	The partial derivative $$\frac{\partial f}{\partial y} = {0.005}\times{3y^2}$$
You can click and drag on the grid below to see the function in 3d-space

The partial derivative $$\frac{\partial f}{\partial x} = {0.005}\times{3x^2}$$