This site is being phased out.
Differentials
Redirect to:
Question: Can we interpret the derivative $\frac{dy}{dx}$ as a fraction?
Emphatic answer: NO!!!
Advanced answer: Yes, here's how.
Suppose we have a function $y=f(x)$ and we are to study it behavior around point $x=a$. Then
So it is a fraction if we create new variables, $dx$ and $dy$.
Here's the idea.
Suppose $y = f(x)$, $f^{\prime}(a) = 3$, or $$ \frac{dy}{dx} = 3. $$ Let's re-write! How about this? $$dy = 3 dx. $$ These are the new variables. What is their meaning?
This is where it comes from:
The slope of the secant line is $m = \frac{\Delta y}{\Delta x}$, or $$ \Delta y = m\cdot\Delta x. $$
What if it's the tangent line instead? $$dy = 3 dx. $$ The slope of tangent is 3. Where are $dx$, $dy$?
These are run, $dx$, and rise, $dy$, of the tangent line.
The more advance view is that $dy = 3 dx$ is a differential form of degree $1$.
Question: Why do we need this?
Answer: What if the Universe is curved?
(Einstein: It is).
Then we need to make a careful distinction between the location, $x$, and the direction, $dx$.
Note: Differentials $dx,dy$ formalize Leibniz’s notion of infinitesimals.