If a certain function has multiple variables, the partial derivatives calculate the rate of change of it concerning one of the variables while holding the other variables fixed or constant. i.e. A partial derivative allows only one variable to change at a time and helps us to analyse surfaces for minimum and maximum points.

For example, if $𝑓(𝑥)=𝑥3$, then we know that $𝑓′(𝑥)=3𝑥2$. What if, $𝑓(𝑥,𝑦)=𝑥3+𝑦2$? Then partial derivatives will produce $𝑓_𝑥’(𝑥,𝑦)=3𝑥2$ and $𝑓_𝑦’(𝑥,𝑦)=2𝑦$. When 𝑓(𝑥,𝑦) is differentiated concerning 𝑥, the 𝑦 component is treated as a constant and vice versa.

Let’s look at an example with 3 variables: $𝑓(𝑥,𝑦,𝑧)=𝑧3−𝑥2𝑦+𝑥𝑦𝑧$

Then, let’s find $f′𝑥,f′𝑦f′𝑥,f𝑦′$ and f′𝑧f𝑧′:

$$𝑓_x’=−2𝑥𝑦+𝑦𝑧$$ $$𝑓_𝑦’=−𝑥^2+𝑥𝑧$$ $$𝑓_𝑧‘=3𝑧^2+𝑥𝑦$$

Another very common notation to represent f′ (derivative) is to use a backwards d(∂).

$$𝑓_𝑥’=\frac{∂𝑓}{∂𝑥}$$

 d(∂) is used to denote partial differential. It is called “dabba”, “del”, “dee”, or “curly dee”.