Page:Calculus Made Easy.pdf/105

$dx$ will be $3$ times as great as $dy$; as magnified in Fig. 21. So, draw the line in Fig. 20 at this slope.

Fig. 21.

(3) Now for a slightly harder case.

Let

$y=ax^{2}+b$.

Again the curve will start on the $y$-axis at a height $b$ above the origin.

Now differentiate. [If you have forgotten, turn back to p. 26; or, rather, don’t turn back, but think out the differentiation.]

${\frac {dy}{dx}}=2ax.$.

Fig. 22.

This shows that the steepness will not be constant: it increases as $x$ increases. At the starting point $P$,