visual_proof_of_chain_rule

A Visual Proof of Calculus Chain Rule Using Python with matplotlib

The Proof:

The tangent of composite function is the composite of the tanget function of each function.

The projection of the tangent of the 3D curve is the tangent of the projection of the 3D curve.

Adding of one more variable corresponds to adding of one more dimension.

Example:

Find the derivative of y with respect to x:

Projection on x-u plane

Projection on u-y plane

Projection on x-y plane

Some discuessions on the meaning of Differentials:

Differential and Derivative Operators are different:

differential of x:

differential of y:

differential of dy i.e. d(dy):

differential of y divided by differential of x:

1st derivative of y with respect to x:

2nd derivative of y with respect to x:

but a lot of literature will write it like this . Which shouldn't be treated as fraction of over

What they are not:

Some existing explanations:

dx is infinitesimal of Δx or very tiny tiny Δx or a number who's absolute value is smaller then all positive real number but not equal to 0. (Somebody make this rigious by inventing standard analysis.) Refer to this youtube video on how the word "small" is defined in mathematics. Differential of a function and its geometrical meaning.
Some define dy as linear approximation of Δy. It can be proved that dy = f'(x) Δx. Following this definition, dx = Δx. That means you have to define x as independent variable and y as dependent variable. That means linear approximation is the tangent line. The tangent line at a point on a curve is the same no matter you treat x or y as dependent variable. But people using this definition usually don't explicity indicate which variable is independent and just imply it by context. And furthermore they will define = 0 as they define dx (=Δx) as constant. So that dy is dependent both on independent variable x and constant dx.
Some say they are symbolic(?).

My explanation:

dx and dy can be defined as the 'Δx' and 'Δy' of the tangent line. dx is not necessarily equal to Δx, they are independent. dx and dy can take large value (but usually we are interested in the case when they take small value). This can be extended to multi variable function as dx, dy, dz... are the 'Δx', 'Δy', 'Δz'... of the linear approximation function. (e.g. for single variable function, the linear approximation function is the tangent line equation. For two variable function, it is the tangent plane equation.) In 1) dx and dy is treated as fundamental concept and slope as derived concept. Here slope is treated as fundamental concept and dx and dy as derived concept.