If we deal with our problem not knowing, or pretending not to know the general theory encompassing the concrete case before us, if we tackle the prob… - George Pólya

<math>\frac {dy}{dx} = \frac {\omega^2x}{g}</math>...The first derivative, the result of the differentiation of <math>y</math> with respect to <math>x</math>, was written by Leibniz in the form
<math>\frac {dy}{dx}</math>...Leibniz's notation ...is both extremely useful and dangerous. Today, as the concepts of limit and derivative are sufficiently clarified, the use of the notation... need not be dangerous. Yet, the situation was different in the 150 years between the discovery of calculus by Newton and Leibniz and the time of Cauchy. The derivative <math>\frac {dy}{dx}</math> was considered as the ratio of two "infinitely small quanitites", of the infinitesimals <math>dy</math> and <math>dx</math>. ...it greatly facilitated the systematization of the rules of the calculus and gave intuitive meaning to its formulas. Yet this consideration was also obscure... it brought mathematics into disrepute... some of the best minds... such as... Berkeley, complained that calculus is incomprehensible. ...<math>\frac {dy}{dx}</math> is the limit of a ratio of <math>dy</math> to <math>dx</math>... Once we have realized this sufficiently clearly, we may, under certain circumstances, treat <math>\frac {dy}{dx}</math> so as if it were a ratio... and multiply by <math>dx</math> to achieve the separation of variables. We get
<math>{dy} = \frac {\omega^2x}{g}xdx</math>

Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.

It is better to solve one problem five different ways, than to solve five problems one way.