When can you divide out $dx$ in an integral as if it is a fraction?

It is a common misconception that $dx$ is an "infinitesimal". $dx$ is a differential, i.e. an arbitrary, nonzero quantity representing a variation.

When there is a dependent variable, say $y=f(x)$, the differential of $y$ is related to that of $x$ by

$$dy=f'(x)\,dx.$$

By this definition, $dy$ is "the linear part of the variation of $y$ for a given variation of $x$", as explained by Taylors' development.

This is to be contrasted with

$$\Delta y=f(x+\Delta x)-f(x)=f'(x)\Delta x+R(x,\Delta x)$$ (where $R$ is a remainder term) which is the ordinary variation.

Hence, $dx,dy$ can really be handled like numbers, and

$$\frac{dy}{dx}=f'(x)=\lim_{\Delta x\to 0}\frac{\Delta y}{\Delta x}.$$

Example:

For the function $y=x^2$,

$$\Delta y=(x+\Delta x)^2-x^2=2x\Delta x+\Delta^2 x$$ while the linear part is

$$dy=2x\,dx.$$

Note that $dx$ and $\Delta x$ both represent arbitrary variations, but I kept both for the symmetry of the formulas.

Tags:

Integration

Calculus

Ordinary Differential Equations

Related

UC Berkeley Integral Problem: Show that $\int_0^{2\pi} \frac{\min(\sin x, \cos x)}{\max(e^{\sin x},e^{\cos x})}\ {\rm d}x = -4\sinh(1/{\sqrt2})$. How to solve this ODE with the Laplace transform? Crux problem #33 with vector approach Argument of Feynman for equivalence of dot product definitions Does there exist a monotonically decreasing function that is its own derivative? Dice: Rolling at least N successes where number of succeses vary by dice value Show the $\arcsin$ identity: $ \arcsin(1 - 2x) + 2\arcsin(\sqrt{x}) = \pi / 2$ Show that $\phi(x):=\sum_{n=1}^{\infty}\frac{(-1)^{n}}{\sqrt{n}(1+\frac{x^{2}}{n})^{n}}$ is differentiable on $\mathbb{R}$. How does the total derivative account for dependencies behind variables (intuitively)? Show that $\sum\limits_{j,k=2}^\infty\frac{1}{j^k}$ converges and calculate the limit of the series Is this proposed method of finding primes valid? If so, would it be effective? Does anyone recall this linear algebra survey of results?

Recent Posts