Page:Calculus Made Easy.pdf/55

CHAPTER VI.

SUMS, DIFFERENCES, PRODUCTS AND QUOTIENTS.

We have learned how to differentiate simple algebraical functions such as $x^{2}+c$ or $ax^{4}$, and we have now to consider how to tackle the sum of two or more functions.

For instance, let

$y=(x^{2}+c)+(ax^{4}+b)$;

what will its ${\frac {dy}{dx}}$ be? How are we to go to work on this new job?

The answer to this question is quite simple: just differentiate them, one after the other, thus:

${\frac {dy}{dx}}=2x+4ax^{3}$. (Ans.)

If you have any doubt whether this is right, try a more general case, working it by first principles. And this is the way.

Let $y=u+v$, where u is any function of $x$, and $v$ any other function of $x$. Then, letting $x$ increase to $x+dx$, $y$ will increase to $y+dy$; and $u$ will increase to $u+du$; and $v$ to $v+dv$.