Derivative as Slope of the Tangent

From secant to tangent. Let $y=f(x)$ be a curve, and fix a point

$A=(a,f(a))$ on it. Choose a second, nearby point $B=(a+h,f(a+h))$ on the same curve, for some

small $h\neq0$. The straight line joining $A$ and $B$ is called a secant line, and its slope

is the familiar rise-over-run ratio:

This is exactly the same difference quotient that appears in the rate-of-change definition of

Section 6 -- here interpreted geometrically rather than physically.

Letting $B$ slide toward $A$. Now let $h\to0$, so that $B$ slides along the curve toward $A$.

As it does, the secant line $AB$ rotates about the fixed point $A$, and (provided the curve is

smooth enough at $A$) approaches a limiting straight-line position. This limiting line is called

the tangent to the curve $y=f(x)$ at the point $A$ -- informally, the straight line that just

"touches" the curve at $A$ without crossing through it locally.

Slope of the tangent = the derivative. Since the slope of the secant is

$\dfrac{f(a+h)-f(a)}{h}$ and the tangent is the limiting position of the secant as $h\to0$, the

slope of the tangent at $A=(a,f(a))$ is

This is precisely the same limit as $f'(a)$, the derivative of $f$ at $x=a$ (formally defined

in Section 8) -- the algebraic definition (instantaneous rate of change) and the geometric

definition (slope of the tangent) are not two different quantities that happen to share a

formula; they are, literally, the exact same limit, viewed two different ways.

Equation of the tangent line. Once the slope $f'(a)$ is known, the tangent line at …