We want the derivative of a product $f \cdot g$, which we can picture as the area of a rectangle with sides $f$ and $g$. Nudging the input by a small amount grows each side by its own differential, $df$ and $dg$:

The new area exceeds the old by two thin strips plus a small corner. The strips have areas $g \,df$ and $f\,dg$, whereas the corner $df\,dg$ is a second order contribution and drops out in the limit. What survives is exactly the product rule:

$$d(fg) = f \, dg + g \,df.$$

As an example, we will now visually prove the power rule

$$dx^n = n x^{n-1} \, dx$$

for the case $n = 3$, but the same argument holds for arbitrary $n$. Take $x^3$ as the volume of a cube with sides $x$, and grow each side by $dx$. Three faces of the cube each grow outward into a slab of volume $x^2\,dx$, so the volume gains $3x^2\,dx$, the derivative of $x^3$. There are some remaining pieces: three long edges of order $x\,dx^2$ and a single corner cube $dx^3$. Those are higher order and vanish in the limit.