Calculus: Pressure on a wall (If the pressure at the center of a volume is given, then the pressure on the wall...)

[I'mLearning]

I'm reading abouot the pressure gradient force.

If the pressure at the center of a volume is given, then the pressure on the wall of the volume may be expressed in a Taylor series expansion:
$$p_{0}+\frac{\partial\,p}{\partial\,x}\frac{\delta\,x}{2}+higher-order terms$$.

$$\delta\,x$$ is the volume width and $$p_{0}$$ is the pressure at the center.

I'd like to understand without getting lost in the maths. What is the essential idea:

 

[blamocur]

Without seeing more context what you posted looks like the standard approximation using derivatives. I.e., if the pressure is a function of $x$, then you can write $p(x_0 + \delta x) \approx p(x_0) + \frac{\partial p}{\partial x} \delta x$.

BTW, the use of partial derivatives implies that the pressure depends on some other variables too.