Page:Calculus Made Easy.pdf/49

Carefully compare the two figures, and verify by inspection that the height of the ordinate of the derived curve, Fig. 6a, is proportional to the slope of the original curve,^[1] Fig. 6, at the corresponding value of $x$. To the left of the origin, where the original curve slopes negatively (that is, downward from left to right) the corresponding ordinates of the derived curve are negative.

Now if we look back at p. 19, we shall see that simply differentiating $x^{2}$ gives us $2x$. So that the differential coefficient of $7x^{2}$ is just $7$ times as big as that of $x^{2}$. If we had taken $8x^{2}$, the differential coefficient would have come out eight times as great as that of $x^{2}$. If we put $y=ax^{2}$, we shall get

${\frac {dy}{dx}}=a\times 2x$.

If we had begun with $y=ax^{n}$, we should have had ${\frac {dy}{dx}}=a\times nx^{n-1}$. So that any mere multiplication by a constant reappears as a mere multiplication when the thing is differentiated. And, what is true about multiplication is equally true about division: for if, in the example above, we had taken as the constant ${\tfrac {1}{7}}$ instead of $7$, we should have had the same ${\tfrac {1}{7}}$ come out in the result after differentiation.

1. See p. 77 about slopes of curves.