Page:Calculus Made Easy.pdf/38

CHAPTER IV.

Simplest Cases.

Now let us see how, on first principles, we can differentiate some simple algebraical expression.

Case 1.

Let us begin with the simple expression $y=x^{2}$. Now remember that the fundamental notion about the calculus is the idea of growing. Mathematicians call it varying. Now as $y$ and $x^{2}$ are equal to one another, it is clear that if $x$ grows, $x^{2}$ will also grow. And if $x^{2}$ grows, then $y$ will also grow. What we have got to find out is the proportion between the growing of $y$ and the growing of $x$. In other words our task is to find out the ratio between $dy$ and $dx$, or, in brief, to find the value of ${\frac {dy}{dx}}$.

Let $x$, then, grow a little bit bigger and become $x+dx$; similarly, $y$ will grow a bit bigger and will become $y+dy$. Then, clearly, it will still be true that the enlarged $y$ will be equal to the square of the enlarged $x$. Writing this down, we have:

$y+dy=(x+dx)^{2}$.

Doing the squaring we get:

$y+dy=x^{2}+2x\cdot dx+(dx)^{2}$.