Why Do We Write Dx?

Take this function:

F(x) = x^2

We all know that the derivative of this function is:

F'(x) = 2x

So when we try to take the antiderivative of F'(x):

S F'(x) dx = S 2x dx = x^2 +C

My question is, why do we need the dx there? Please give me a reason other than "notation". Because my teacher basically marked points off for every time I showed my work as:

S 2x = x^2 +C

dx is what you're integrating over. x is the variable of integration.

But why is S 2x incorrect?

Because you aren't integrating over anything. it could be S 2x dy, which would give you a very different result.

What if you have dy = x, and you want to find y?

dy = x

S dy = S x

y = S x ?

What if you have dy = x, and you want to find y?

dy = x

S dy = S x

y = S x ?

If
S dy = S x, then
S x = y + C

dx is an independent variable, and it's called a differential of a function.



Say you have a function f(x), and:
a + Δx = b
f(a) + Δy = f(b).

The relationship between Δx and Δy is that if x increases by Δx (at a particular point), then y increases by Δy. The ratio Δy/Δx is the slope between the two points on the curve.

dx and dy are basically the same as Δx and Δy, except now you're working with the tangent line instead of the curve itself. For a given value of dx, if x increases by dx, then the tangent line increases by dy. Since it's a line and has constant slope, the value of dx has no bearing on the ratio dy/dx, which is always equal to the slope of the tangent line.

In the context of integration, remember that an integral is a limit of a Reimann sum (areas of rectangles). Δx in this case corresponds to the width of a rectangle. When you take the limit, Δx goes to 0, and by some consequence of definition, it gets denoted by dx which is the same idea as above. I really don't know why this is.

Sorry my explanation isn't very clear; calculus courses tend to skip this topic and I only remember some of this from when I read the section in the textbook once.

In the context of integration, remember that an integral is a limit of a Reimann sum (areas of rectangles). Δx in this case corresponds to the width of a rectangle. When you take the limit, Δx goes to 0, and by some consequence of definition, it gets denoted by dx which is the same idea as above. I really don't know why this is.

d is used to denote an infinititesimally small delta. Integrals can only be taken on differentials, you cannot integrate a delta x, or an x. That doesn't make sense.