>>4447 If the terms "total derivative" and "partial derivative" mean nothing to you, then tl;dr yes.
If you're looking for a total derivative, and a is a constant, then the partial derivative in >>4448 is also total. If a is invariant in x but not constant, then it'll only be partial and needs the del (∂) rather than the d.
By way of intuition, this is how I'd imagine it, though it's been years since I did calculus in anger so it could be horribly wrong.
Suppose you have a cube of width x. Increase one dimension by dx and the volume will increase by (x^2)dx. Doing this on on all three dimensions will increase the volume by 3(x^2)dx. Integrate and we get x^3 as the formula for volume. Is this right?
More examples.
The surface area of a sphere is 4pi(r^2). An increase in radius by dr will increase the volume by 4pi(r^2)dr. Integrate and we get (4/3)pi(r^3) as the formula for volume.
The circumference of a circle is (2pi)r. Integrate and we get area = pi(r^2).
The volume of the cross-section of an n-dimensional cone is kw^(n-1), where w is the width and k some constant. Integrate this over the length of the cone to give V = kAl/n where A is the volume of the base and l the length.
The case k=1, A=h, n=2 gives hl/2 as the area of a triangle.
