The Derivative and Tangent Line Problem

The tangent line $t$ (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Informally, it is a line through a pair of infinitely close points on the curve. More precisely, a straight line is said to be a tangent of a curve $y = f(x)$ at a point $x = c$ on the curve if the line passes through the point $(c, f(c))$ on the curve and has slope $f'(c)$ where f' is the derivative of $f$.

Tangent to a Curve

The line shows the tangent to the curve at the point represented by the dot. It barely touches the curve and shows the rate of change slope at the point.

Suppose that a curve is given as the graph of a function, $y = f(x)$. To find the tangent line at the point $p = (a, f(a))$, consider another nearby point $q = (a + h, f(a + h))$ on the curve. The slope of the secant line passing through $p$ and $q$ is equal to the difference quotient

$\displaystyle{\frac{f(a + h) - f(a)}{h}}$.

As the point $q$ approaches $p$, which corresponds to making $h$ smaller and smaller, the difference quotient should approach a certain limiting value $k$, which is the slope of the tangent line at the point $p$. If $k$ is known, the equation of the tangent line can be found in the point-slope form:

$y - f(a) = k(x-a)$

Suppose that the graph does not have a break or a sharp edge at $p$ and it is neither vertical nor too wiggly near $p$. Then there is a unique value of $k$ such that, as $h$approaches $0$, the difference quotient gets closer and closer to $k$, and the distance between them becomes negligible compared with the size of $h$, if $h$ is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function $f$. This limit is the derivative of the function $f$ at $x=a$, denoted $f'(a)$. Using derivatives, the equation of the tangent line can be stated as follows:

$y = f(a) + f{(a)}'(x-a)$.