Slopes, Tangent Lines, and the Derivative

image of a secant line derivative on graph Defining and finding the slope of a curve at a point are often referred to as key ideas of calculus. It is from the slope of a line that it is possible to find the slope of a curve, and from the slope of a curve it is possible to find a tangent line to the curve. It is the slope of the tangent line at a point on the curve that forms the basis of the derivative concept.

Slope of the Secant Line

Approximating the slope of a curve at a point P on the curve can be accomplished by taking a point P and another point Q on the curve that is relatively close to P and computing the slope of the line connecting them. When any two points on a curve are connected, a secant line is produced. Given P(x, f(x)) and nearby point Q(a, f(a)), the slope of the secant line is denoted by equation image indicator $LaTeX: m_{\sec}=\frac{\Delta y}{\Delta x}=\frac{f\left(x\right)-f\left(a\right)}{x-a}$ . Another expression for the slope of a secant line can be derived by letting h = x - a, which means x = a + h. The slope of the secant line above then becomes $LaTeX: m_{\sec}=\frac{f\left(a+h\right)-f\left(a\right)}{h}$ . This ratio equation image indicator $LaTeX: \frac{f\left(a+h\right)-f\left(a\right)}{h}$ is called the difference quotient. It measures the average rate of change of the function f and can be interpreted as the slope of a secant line.

View the presentation below from the beginning to 12:33.

Tangency to a Curve

One of the classic problems in calculus is the tangent line problem. This problem has been studied by many mathematicians since Archimedes explored the question in Antiquity.

Defining Slopes and Tangent Lines

image of curvy road into tunnel A line that touches a curve at a point and has the same slope as the curve at the point is a tangent line. In practical terms imagine that it is dark outside and you are looking down out of an airplane window at a curvy highway. You see a car's headlights following the highway and notice that when the car reaches a certain point on one of the road's curves, the headlights point alongside the road in a straight path rather than along the curvy road. You realize that the car has just driven off the road in the direction of a tangent line.

image of a tangent line on a graph

A tangent line to a curve can be thought of as the limiting position of the secant line. More formally, the tangent line to the curve y = f(x) at the point P(a, f(a)) is the line through P with slope equation image indicator $LaTeX: m=\lim_{x\to a}\frac{f\left(x\right)-f\left(a\right)}{x-a}$ , provided this limit exists. As with the slope of the secant line, another expression for the slope of a tangent line can be derived by letting h = x - a, which means x = a + h. Notice that as x -> a, x - a -> 0 and h -> 0. Then the slope of the tangent line above becomes equation image indicator $LaTeX: m=\lim_{h\to 0}\frac{f\left(a+h\right)-f\left(a\right)}{a}$ . Since the slope of the tangent line depends on both the function f and the number a, we indicate this by writing $LaTeX: m=f'\left(a\right)=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$ . When we replace a by the more generic x, we are defining a new function equation image indicator $LaTeX: m=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$ .

View the presentation below on how a secant line becomes a tangent line.

Derivative of a Function

Definition of Derivative

The derivative of f is the function f'(x) whose value at x is defined by equation image indicator $LaTeX: f'(x)=\lim_{h\to0}\frac{f\left(x+h\right)-f\left(x\right)}{h}$ , provided this limit exists. The limit used to define the slope of a tangent line is also used to define differentiation, the process of determining the derivative of a function. If we interpret the difference quotient as an average rate of change, the derivative (limit of the difference quotient) gives the instantaneous rate of change of the function with respect to x at x= a. Thus the derivative can be interpreted three ways: a) the slope of the tangent line to a curve at a point, b) the slope of a curve at a point, and c) the instantaneous rate of change of a curve at a point.

Derivative Notation

Did you know? The "prime" notations of y and y² come from notations that Newton used for derivatives. The d/dx notations are similar to those used by Leiloniz. The most common notations for the derivative of a function y = f(x) are represented in the table below.

Derivative Notations
Notation	Read As	Type
y'	"y prime"	Prime Notation
	"f prime of x"	Prime Notation
	"dy dx" or "the derivative of y with respect to x"	Leibniz Notation and Differential Notation
	"df dx" or "the derivative of f with respect to x"	Variation of Leibniz Notation and Differential Notation
	"ddx of f" or as "the derivative of f with respect to x"	Variation of Leibniz Notation and Differential Notation
	"D sub x of f"	Operator Notation
	"D of f of x"	Operator Notation

equation image indicator $LaTeX: \frac{dy}{dx}\biggr|_{x=a}$ is the Leibniz/differential notation used to evaluate a derivative at a given value a of the independent variable. The prime notation for evaluating a derivative at a given value a is f'(a).

Equations of Tangent Lines

Recall that writing an equation of a tangent line requires one point on the curve (point of tangency), the slope of the tangent line (the derivative of the function at the point of tangency), and the point-slope formula, y - y₁ = m_T(x - x₁).

Many real-world situations use the equation of a tangent line to a curve to predict either past, present, or future behavior when no data is available for the time in question. CLICK HERE to view the presentation below related to average yearly per capita health care expenditures ($) for various years since 1960.

Watch the video below to Find the Equation of a Tangent Line Using the Product Rule

Graphing the Derivative of a Function

Drawing tangent lines to the graph of f enables us to estimate f'(x) for various values of x. By plotting the points (x, f'(x) and connecting them with a smooth curve, a reasonable plot of y = f'(x) results. From the graph of y = f'(x) it is possible to see

where the rate of change of f is positive, negative, or zero;
the size of the growth rate at any x and its size in relation to the size of f(x); and
where the rate of change is increasing or decreasing.

View the presentation below relating the graph of a function to the graph of its derivative.

Given any function graph, it is possible to predict the graph of its derivative based on plotting estimates of the slopes of tangents drawn to the curve. View below how this can be accomplished.

Differentiability and Continuity

Differentiable versus Non-Differentiable Functions

Functions are described as differentiable if a derivative exists at every point in the domain of f. Many common functions such as polynomials, rational, exponential, and trigonometric are differentiable, as are composites, sums, differences, products, powers, and quotients of differentiable functions, where defined.

A function is not differentiable at a if the limit equation image indicator $LaTeX: \lim_{h \to 0}\frac{f(a+h)-f(a)}{h}$ fails to exist. Functions may be non-differentiable at a point if one of the following situations occurs.

situation 1 graph on calculator

Situation - Non-differentiability situation: The graph of a function f has a "corner" (left and right limits differ).

situation 2 graph on calculator

Situation - Non-differentiability situation: The graph of function f has a "cusp" (left and right limits approach infinity from one side and negative infinity from the other side).

situation 3 graph on calculator

Situation - Non-differentiability situation: The function has a vertical tangent at a point (left and right limits approach infinity from both sides or negative infinity from both sides.)

situation 4 graph on calculator: discontinuity

Situation - Non-differentiability situation: The function is discontinuous at a point. (This function is discontinuous at every integer point.)

Vertical Tangents and No Tangents

The definition of a tangent line to a curve does not cover the possibility of a vertical tangent line. For vertical tangent lines, the following definition is used. If f is continuous at c and equation image indicator $LaTeX: \lim_{\Delta x \to 0}\frac{f(c+\Delta x) -f(c )}{\Delta x}=\infty$ or $LaTeX: \lim_{\Delta x \to 0}\frac{f(c+\Delta x) -f(c )}{\Delta x}=-\infty$ , the vertical line x = c passing through (c, f(c)) is a vertical tangent line to the graph of f.

Differentiability Implies Continuity

If f has a derivative at x = c, then f is continuous at x = c. Note that the converse is not true. The function f(x) = |x| is continuous at 0 since equation image indicator $LaTeX: \lim_{x \to 0}f\left(x\right)=\lim_{x \to 0}\left|x\right|=0=f\left(0\right)$ . However, f(x) is not differentiable since $LaTeX: \lim_{x \to 0^-}|x|=-1\: and \: \lim_{x \to 0^+}|x|=1, \lim_{x \to 0}|x|$ does not exist. In general, a function need not have a derivative at a point where it is continuous.

Numerical Derivative

Some graphing utilities can compute derivatives symbolically by using differentiation rules, and others approximate derivatives by applying a numerical method. A numerical derivative is a method for approximating a derivative of a function f at a specific point that utilizes the symmetric difference quotient. Typically, graphing calculators use the symmetric difference quotient equation image indicator $LaTeX: \frac{f\left(a+h\right)-f\left(a-h\right)}{2h}$ to determine the numerical derivative at (a, f(a)). It should be noted that the symmetric difference quotient uses two points whose x-coordinates, a - h and a + h, are symmetrically positioned on either side of a. For a function f, the numerical derivative at a point (a, f(a)) can be denoted as NDER (f(x), a) or nDeriv(f(x), a).

A calculator calculus command that returns the approximate value of a derivative with respect to a given variable at a stated value of x is nDeriv(expression, variable, value). Although the nDeriv command generally produces accurate approximations, care should be taken when using the numerical derivative since it is possible for the calculator command to return an incorrect result. Determining whether or not to accept the nDeriv result requires a thorough understanding of the graphs of non-differentiable functions. Review how to use the nDeriv graphing calculator command to produce a graph below.

Consider the situation where a graphing calculator produced equation image indicator $LaTeX: nDeriv\left(\sqrt[3]{x},x,0\right)=100$ but the graphs of $LaTeX: y=\sqrt[3]{x}$ and y' suggest correctly that y' does not exist at x = 0.

image of math menu on calculator image of y = on calculator image of derivative graph on calculator

Slopes, Tangent Lines, and the Derivative Practice

Slopes, Tangent Lines, and the Derivative: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of slopes, tangent lines, and the derivative.

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