Intereting Posts

Applications of Pseudodifferential Operators
Deriving equations of motion in spherical coordinates
Inequality with condition $x^2+y^2+z^2=1$.
Trace and the coefficients of the characteristic polynomial of a matrix
Assume that $f$ is uniformly continuous. Prove that $\lim_{x→∞} f(x) = 0.$
Grandi's series contradiction
Counterexample Of Banach Fixed Point (Banach's Contraction) Theorem
Factor $x^{14}+8x^{13}+3$ over the rationals
An alternative way to calculate $\log(x)$
Prove that $1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+…=\sqrt{3}$
Prove the inequality $|xy|\leq\frac{1}{2}(x^2+y^2)$
Show $\sum_n \frac{z^{2^n}}{1-z^{2^{n+1}}} = \frac{z}{1-z}$
Understanding Primitive Polynomials in GF(2)?
Markov property question
How to find PV $\int_0^\infty \frac{\log \cos^2 \alpha x}{\beta^2-x^2} \, \mathrm dx=\alpha \pi$

I have been forewarned about it, I have read the answers here, but I haven’t seen a counter example where it doesn’t work. I know that it isnt really a fraction, but does it effectively get the same result in all cases, or are there counterexamples I must be warned about.

I do not mean obvious counter examples, like $$\frac{dy}{dx}+\frac{du}{dv} = \frac{dydv+dxdu}{dx dv}$$ which as far as I know doesn’t really mean anything.

- Motivating infinite series
- How to integrate this improper integral.
- Proof for transformation of random variables formula
- About matrix derivative
- Other challenging logarithmic integral $\int_0^1 \frac{\log^2(x)\log(1-x)\log(1+x)}{x}dx$
- Calculate $\lim_{n\rightarrow \infty} \frac{1}{n}\sum_{k=1}^{n}k\sin\left(\frac{a}{k}\right)$

- The Limit of $x\left(\sqrt{a}-1\right)$ as $x\to\infty$.
- Limits of the solutions to $x\sin x = 1$
- How to prove $f(x)=ax$ if $f(x+y)=f(x)+f(y)$ and $f$ is locally integrable
- How to derive this interesting identity for $\log(\sin(x))$
- Proving the continued fraction representation of $\sqrt{2}$
- Calculate a limit $\lim_{x \to \pi/2} \frac{\sqrt{ \sin x} - \sqrt{ \sin x}}{\cos^2x}$
- Is a smooth function characterized by its value on any (non-empty) open interval?
- Solving higher order logarithms integrals without the beta function
- Finding the value of $3(\alpha-\beta)^2$ if $\int_0^2 f(x)dx=f(\alpha) +f(\beta)$ for all $f$
- If $f(\frac{x}{y})=\frac{f(x)}{f(y)} \, , f(y),y \neq 0$ and $f'(1)=2$ then $f(x)=$?

$\frac{dy}{dx}$ can indeed be thought of as a ratio. The question is, what do $dx$ and $dy$ represent in this ratio? The answer is, changes in $x$ and $y$ *along the tangent line* to the curve at the point in question, rather than along the curve itself. See e.g. https://www.encyclopediaofmath.org/index.php/Differential

The chain rule in two dimensions is counter-intuitive at first if you think of the derivative (and partial derivatives) as ratios:

$$

\frac{\mathrm d f}{\mathrm dt} = \frac{\partial f}{\partial x} \frac{\mathrm dx}{\mathrm dt} + \frac{\partial f}{\partial y} \frac{\mathrm dy}{\mathrm dt}.

$$

Here, $f$ is a function of two variables, $x$ and $y$, both of which are functions of $t$.

This question has come up at MathOverflow:

https://mathoverflow.net/questions/73492/how-misleading-is-it-to-regard-fracdydx-as-a-fraction

and overlaps with other questions on this site:

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

If $\frac{dy}{dt}dt$ doesn't cancel, then what do you call it?

There are several interesting answers at each of those links.

The rule of thumb to use is to only think in this manner when there is only one independent variable (e.g. when all of your variables are to be thought of as functions of $x$). When all of the variables you are using depend on each other, there a variety of (fully rigorous) ways to think of expressions like “$dx$” as simply being a value with funny units, which cancel out in “$\frac{dy}{dx}$”, resulting in a perfectly ordinary value.

But when there are more variables (e.g. when considering $z = f(x,y)$), this way of thinking no longer works. (Or more accurately, it does still work, but $dx, dy, dz$ behave like vectors so division doesn’t make sense)

- Topology of the power set
- Solving $\sin\theta -1=\cos\theta $
- Sum of derangements and binomial coefficients
- Congruent Polynomials
- Open Mapping Theorem: counterexample
- Solving $(f(x))^2 = f(\sqrt{2}x)$
- Bijection between natural numbers $\mathbb{N}$ and natural plane $\mathbb{N} \times \mathbb{N}$
- If $n$ is an integer then $\gcd(2n+3,3n-2)=1\text{ or ?} $
- Why do remainders show cyclic pattern?
- Finding a basis for the field extension $\mathbb{Q}(\sqrt{2}+\sqrt{4})$
- Why is integration so much harder than differentiation?
- Showing that $\sum\limits_{n \text{ odd}}\frac{1}{n\sinh\pi n}=\frac{\ln 2}{8}$
- Find solution of equation $(z+1)^5=z^5$
- Finiteness of the dimension of a normed space and compactness
- Manifold with 3 nondegenerate critical points