Now that we have defined limits and have learned techniques for computing them, we visit the problems of finding tangent lines and velocities. The special type of limit that occurs in both of these problems is called a derivative and we will see that it can be interpreted as a rate of change in any of the natural or social sciences or engineering.
If a curve $C$ has equation $y = f(x)$ and we want to find the tangent line to $C$ at the point $P(a,f(a))$, then we consider a nearby point $Q(x,f(x))$, where $x\ne a$, and compute the slope of the secant line $PQ$:
$$ m_{PQ}=\frac{f(x)-f(a)}{x-a} $$
Then we let $Q$ approach $P$ along the curve $C$ by letting $x$ approach $a$. If $m_{PQ}$ approaches a number $m$, then we define the tangent line $l$, to be the line through $P$ with slope $m$.
The tangent line to the curve $y=f(x)$ at the point $P(a,f(a))$ is the line through $P$ with slope
$$ m=\lim_{x\rarr a}\frac{f(x)-f(a)}{x-a} $$
provided that this limit exists.
We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point. The idea is that if we zoom in far enough toward the point, the curve looks almost like a straight line.
The more we zoom in, the more the parabola looks like a line. In other words, the curve becomes almost indistinguishable from its tangent line.
<aside>
Find an equation of the tangent line to the parabola $y=x^2$ at the point $P(1,1)$.
</aside>
<aside>
Find an equation of the tangent line to the hyperbola $y=3/x$ at the point $(3,1)$.
</aside>
We have seen that the same type of limit arises in finding the slope of a tangent line. In fact, limits of the form
$$ \lim_{h\rarr0}\frac{f(a+h)-f(a)}{h} $$
arise whenever we calculate a rate of change in any of the sciences or engineering, such as a rate of reaction in chemistry or a marginal cost in economics. Since this type of limit occurs so widely, it is given a special name and notation.
The derivative of a function $f$ at a number $a$, denoted by $f'(a)$, is
$$ f'(a)=\lim_{h\rarr0}\frac{f(a+h)-f(a)}{h} $$
If we write $x=a+h$, then we have $h=x-a$ and $h$ approaches $0$ if and only if $x$ approaches $a$. Therefore an equivalent way of stating the definition of the derivative, as we saw in finding tangent lines is