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.



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



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