The derivative and the tangent line

A line touching a curve at a single point is a tangent¹. Once the derivative is known, the equation of the tangent follows at once. We find the tangent to $f(x) = x^3 - 3x$ at $x = 2$ .

The procedure

Step	Computation	Result
Find the derivative	$f'(x) = 3x^2 - 3$	a formula
Compute the slope	$f'(2) = 12 - 3$	$9$
Compute the point	$f(2) = 8 - 6$	$2$

The line of slope $9$ through $(2, 2)$ is therefore as follows.

y = f(2) + f'(2)(x - 2) = 2 + 9(x - 2) = 9x - 16

In general the tangent at $x = a$ has the following form.

y = f(a) + f'(a)(x - a)

Whatever the curve, the order of the steps does not change.

A tangent may cross the curve

A tangent touches the curve at its point of tangency, but it may perfectly well cross the curve somewhere else. Solving $x^3 - 3x = 9x - 16$ shows this.

x^3 - 12x + 16 = (x - 2)^2(x + 4) = 0

Here $x = 2$ is a double root and $x = -4$ is another. The tangent cuts through the curve at $(-4, -52)$ .

That the point of tangency gives a double root is the algebraic meaning of touching. The test of solving a curve and a line together and looking for a double root comes from exactly this.

A tangent is also a linear approximation

The value $f'(2) = 9$ says that near $x = 2$ , a small increase in $x$ raises $y$ by about nine times as much.

$x$	$f(x)$	Value on the tangent
$2$	$2$	$2$
$2.1$	$2.961$	$2.9$

The two are close. A tangent is also the linear approximation of the curve.

The normal, and horizontal tangents

The line perpendicular to the tangent at the point of tangency is the normal, and its slope is $-\dfrac{1}{f'(a)} = -\dfrac{1}{9}$ .

A tangent is horizontal when $f'(a) = 0$ , which here means $3x^2 - 3 = 0$ , that is $x = \pm 1$ : the local maximum and the local minimum. Examining the slope of the tangent is exactly the same as examining where the function rises and falls.

The cubic on the graph is $y = x^3 - 3x$ , the line is the tangent $y = 9x - 16$ , and the large dot is the point of tangency $(2, 2)$ .

Tangent, Wikipedia