Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/56

Fourth Step.	$\lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}$	$=\lim _{\Delta x\to 0}{\frac {f(x+\Delta x)-f(x)}{\Delta x}}$
(B)		$={\frac {dy}{dx}}=$ value of the derivative at $P$.

But when we let $\Delta x{\dot {=}}0$, the point $Q$ will move along the curve and approach nearer and nearer to $P$, the secant will turn about $P$ and approach the tangent as a limiting position, and we have also

	$\lim _{\Delta x\to 0}{\frac {\Delta y}{\Delta x}}$	$=\lim _{\Delta x\to 0}\tan \phi =\tan \tau$
(C)		= slope of the tangent at $P$.

Hence from (B) and (C), ${\tfrac {dy}{dx}}=$ slope of the tangent line $PT$. Therefore

Theorem. The value of the derivative at any point of a curve is equal to the slope of the line drawn tangent to the curve at that point.

It was this tangent problem that led Leibnitz^[1] to the discovery of the Differential Calculus.

Illustrative Example 1. Find the slopes of the tangents to the parabola $y=x^{2}$ at the vertex, and at the point where $x={\tfrac {1}{2}}$.

Solution. Differentiating by General Rule, §31, we get

(A)	${\frac {dy}{dx}}=2x=$ slope of tangent line at any point on curve.

Tangent to a parabola.

To find slope of tangent at vertex, substitute $x=0$ in (A), giving

${\frac {dy}{dx}}=0$.

Therefore the tangent at vertex has the slope zero; that is, it is parallel to the axis of x and in this case coincides with it.

To find slope of tangent at the point $P$, where $x={\tfrac {1}{2}}$, substitute in (A), giving

${\frac {dy}{dx}}=1$;

that is, the tangent at the point $P$ makes an angle of 45° with the axis of $x$.

1. Gottfried Wilhelm Leibnitz (1646-1716) was a native of Leipzig. His remarkable abilities were shown by original investigations in several branches of learning. He was first to publish his discoveries in Calculus in a short essay appearing in the periodical Acta Eruditorum at Leipzig in 1684. It is known, however, that manuscripts on Fluxions written by Newton were already in existence, and from these some claim Leibnitz got the new ideas. The decision of modern times seems to be that both Newton and Leibnitz invented the Calculus independently of each other. The notation nsed to-day was introduced by Leibnitz. See frontispiece.