Tangent Lines and the Derivative at a Point

What is a tangent line to a curve?

A tangent line to a curve at a point is the straight line that fits the curve best at that point: it passes through the point and points in the same direction as the curve there. Close to the point the curve and its tangent line are almost the same thing, which is what the zoom tool below shows: the closer you look, the straighter the curve becomes, until you cannot tell it from its tangent.

The idea comes from circles, where a tangent touches the circle at one point and is perpendicular to the radius (see chords and tangents). For a general curve "touches at exactly one point" is not the right definition, for three reasons:

• A tangent can meet the curve again. The line y=-2 is the tangent to y=x^3-3x at x=1, and it meets the curve a second time at x=-2.
• A line that meets a curve once need not be a tangent. The vertical line x=1 meets the parabola y=x^2 only once, yet the parabola at (1,1) climbs with slope 2; its tangent there is y=2x-1.
• A tangent can cross the curve. At the inflection point x=0 of y=x^3-3x the tangent y=-3x passes from one side of the curve to the other.

So the tangent is a local idea. It says which way the curve is heading at the point, not how the line meets the curve far away. To make it precise we need two steps: the slope of a secant line, which joins two points of the curve, and then what happens to that slope when the two points merge. That is the next section.

Try it: pick a function and a point, then zoom in.

How do you find the slope of a tangent line?

The slope of the tangent line at x=a is the limit of the slopes of the secant lines through (a,f(a)) and a second point (a+h,f(a+h)), as h shrinks toward 0. A secant line is a line through two points of a curve. Its slope is rise over run, the difference quotient: (f(a+h)-f(a))/(h). Take f(x)=x^2 at a=1. Expanding the bracket gives f(1+h)-f(1)=(1+h)^2-1=2h+h^2, and taking out the common factor h gives ((1+h)^2-1)/(h)=(h(2+h))/(h)=2+h (h\ne0). Now let the second point approach (1,1):

• From the right: h=1 gives slope 3, h=0.1 gives 2.1, h=0.01 gives 2.01, h=0.001 gives 2.001.
• From the left: h=-0.1 gives 1.9, h=-0.01 gives 1.99, h=-0.001 gives 1.999.

The slopes close in on 2 from both sides, and we write \lim_h→0(2+h)=2. That limit is the slope of the tangent line.

Why not simply put h=0 in the quotient? Because it gives \frac00, which has no value: the two points would be the same point and there is no line through "two" points. The limit does not ask for the value at h=0; it asks what the quotient approaches when h is near 0 but not 0. After cancelling h the answer is plain.

Two details matter. The limit has to exist from both sides, because the second point may lie to the left (h<0) or to the right (h>0). And when it exists it gives the definition:

The tangent line to the graph of f at P=(a,f(a)) is the line through P with slope m=\lim_h→0(f(a+h)-f(a))/(h), provided this limit exists. If the quotient grows without bound the line is vertical, which is covered below.

What is the derivative of a function at a point?

The derivative of f at a, written f'(a), is the limit of the difference quotient, f'(a)=\lim_h→0(f(a+h)-f(a))/(h), when that limit exists; it is the slope of the tangent line at x=a and the instantaneous rate of change of f there. Writing x=a+h gives the same limit in a second form, f'(a)=\lim_x→ a(f(x)-f(a))/(x-a), which is often handier when the function has a root in it. The derivative at a point is a number. Other notations for it are (df)/(dx)\bigg|_x=a and (dy)/(dx)\bigg|_x=a (Leibniz). A function is differentiable at a when the limit exists.

Three examples show the three usual algebra tricks.

1. A polynomial: expand and cancel h. For f(x)=x^3-3x at a=2 we have f(2)=2, and (2+h)^3=8+12h+6h^2+h^3, so subtracting f(2) leaves (f(2+h)-f(2))/(h)=(9h+6h^2+h^3)/(h). Cancelling h gives 9+6h+h^2, which tends to 9 as h→0. So f'(2)=9, and the tangent line there is steep.

2. A fraction: combine over a common denominator. For f(x)=\frac1x at a=2, using adding fractions, \frac\frac12+h-\frac12h=(2-(2+h))/(2(2+h) h)=(-1)/(2(2+h))→-\frac14. The same steps at any a\ne0 give f'(a)=-(1)/(a^2): the slope is negative everywhere, steep near x=0 and flattening out far away, and it is -\frac14 at both a=2 and a=-2, so there are two tangents of that slope.

3. A root: multiply by the conjugate. For f(x)=√ x at a=4, by rationalizing the numerator: multiplying top and bottom by √(4+h)+2 turns the numerator into (4+h)-4=h, so (√(4+h)-2)/(h)=(h)/(h\left(√(4+h)+2\right)). Cancelling h leaves (1)/(√(4+h)+2), which tends to \frac14 as h→0.

The quotient can be handled from one side at a time. The right-hand derivative uses h→0^+ and the left-hand derivative uses h→0^-; f is differentiable at a exactly when both exist and are equal. Doing the same limit at every a at once gives a new function, the derivative function f', and its rules are shortcuts for this limit; the definition above is what they are shortcuts for.

How do you write the equation of a tangent line?

The tangent line to y=f(x) at x=a is y=f(a)+f'(a)(x-a), the point-slope form of the line through (a,f(a)) with slope f'(a). Three steps: find the height f(a), find the slope f'(a) from the limit, and put both into the formula. Three examples:

• f(x)=x^2 at a=1: f(1)=1 and f'(1)=2, so y=1+2(x-1)=2x-1.
• f(x)=\frac1x at a=2: f(2)=\frac12 and f'(2)=-\frac14, so y=\frac12-\frac14(x-2)=-\frac x4+1.
• f(x)=x^3-3x at a=1: f(1)=-2 and f'(1)=0, so y=-2, a horizontal tangent (slope 0).

The normal line at the point is the line through it at right angles to the tangent. Its slope is -(1)/(f'(a)) when f'(a)\ne0, since the slopes of perpendicular lines multiply to -1; when the tangent is horizontal the normal is the vertical line x=a. For y=x^2 at (1,1) the normal is y=1-\frac12(x-1)=-\frac x2+\frac32.

The tangent line also has an algebraic signature that does not mention limits. Subtract the tangent line from the function: the difference f(x)-\big(f(a)+f'(a)(x-a)\big) is zero at x=a, and for a polynomial it has a double root there. For x^3-3x at a=1 the difference is x^3-3x+2=(x-1)^2(x+2): the double root at 1 is the tangency, and the simple root at x=-2 is the second place where the tangent meets the curve. At the inflection point a=0 the difference is x^3, a triple root, which is why the tangent crosses the curve there.

The same idea finds tangents through an outside point. To draw tangents to y=x^2 from (1,-3), ask for which a the tangent at a passes through that point: a^2+2a(1-a)=-3, which simplifies to a^2-2a-3=0, so (a-3)(a+1)=0. The tangent lines touch the parabola at a=3 and a=-1: y=6x-9 and y=-2x-1. (Both pass through (1,-3): 6-9=-3 and -2-1=-3.)

A tangent line is also an approximation. Near a, f(x)≈ f(a)+f'(a)(x-a). With f(x)=√ x, a=4 and slope \frac14, we get √(4.1)≈2+\frac14·0.1=2.025; the true value is 2.02485…, so the error is under 0.0002. This is one of the ways of approximating an irrational number.

Try it: type a polynomial and a point, or a point outside the curve. Every number is an exact fraction.

What does the derivative at a point mean in real life?

The derivative at a point is the instantaneous rate of change: how fast the output is changing per unit of input at that exact moment, in units of y per unit of x. The difference quotient is the average rate of change over a step of size h, and the derivative is what the average settles on as the step shrinks to nothing.

Speed. Let a ball dropped from a tower fall s(t)=4.9t^2 meters in t seconds. Its average speed from t=1 to t=1+h is (4.9(1+h)^2-4.9)/(h)=9.8+4.9h. Over a whole second (h=1) that is 14.7 m/s, over half a second 12.25 m/s, over 0.1 s it is 10.29 m/s, over 0.01 s it is 9.849 m/s, and as h→0 it tends to 9.8 m/s. That number, s'(1)=9.8 m/s, is the reading on the speedometer at t=1: an average needs a time interval, a speed at an instant needs the limit.

Cost. If making q items costs C(q)=2q^2+30q+400 dollars, then C(11)-C(10)=972-900=72 dollars is the extra cost of the 11th item, which is a difference quotient with h=1. The derivative C'(10)=4·10+30=70 is the marginal cost at 10 items: 70 dollars per additional item, close to the exact extra of 72 dollars. Economists use the derivative as a quick estimate of the cost of one more.

Reading the sign. If f'(a)>0 the graph is rising at a, if f'(a)<0 it is falling, and if f'(a)=0 the tangent is horizontal. A horizontal tangent can be a top, a bottom, or neither: y=x^3 has f'(0)=0 but keeps rising through 0.

Constant rate. For a straight line the rate is the same everywhere, and the tangent line is the line itself. The equal steps of an arithmetic sequence are the same thing in discrete form: a constant difference is a constant slope. A curve has a different slope at each point, and the derivative is what lets you ask for it one point at a time.

Units. The derivative has the units of y divided by the units of x: meters per second for distance against time, dollars per item for cost against quantity. A road that climbs 8 meters for each 100 meters along the ground has slope 0.08, an 8% grade.

Motion along a line (velocity, speed, acceleration, jerk, free fall) and marginal cost and revenue are worked out in full in the derivative as a rate of change.

When does a function have no derivative at a point?

A function has no derivative at a point when the limit of the difference quotient does not exist, which happens at a corner, a cusp, a jump, a vertical tangent, or where the graph wiggles without settling down. The widget below shows each case at x=0 with the slopes of secants from the left and from the right. Here is what to look for:

• A corner. For f(x)=|x| the secant slopes are -1 from the left and +1 from the right. They disagree, so there is no tangent line, and no amount of zoom straightens a corner.
• A cusp. For f(x)=x^2/3 the slopes from the right grow to +∞ and from the left to -∞: two vertical directions that point opposite ways.
• A jump. A function that jumps is not even continuous there, and the difference quotient blows up on at least one side.
• A vertical tangent. For f(x)=x^1/3 the slopes grow to +∞ from both sides: the tangent line is the vertical line x=0. The graph has a tangent line, but its slope is not a number, so f'(0) does not exist.
• Endless wiggling. For f(x)=x\sin\frac1x the secant slope is \sin\frac1h, which keeps swinging between -1 and 1 and never settles.

Not every graph that bends at a point has a corner, either. f(x)=x|x| looks as if it might kink at 0, but its slopes tend to 0 from both sides, so f'(0)=0 and the tangent is horizontal.

Differentiable implies continuous. If f'(a) exists then f(a+h)-f(a)=(f(a+h)-f(a))/(h)· h, which tends to f'(a)·0=0, so f(a+h)→ f(a). That rules out every jump. The converse is false: |x| is continuous everywhere and not differentiable at 0. So continuity is necessary for a derivative but not sufficient.

A practical test follows from the zoom: a graph that is differentiable at a point looks like a straight line when you zoom in on that point, and one that is not does not.

How do you compute a derivative at a point in code?

In code, estimate the derivative with a difference quotient and a small step, preferably the central difference (f(a+h)-f(a-h))/(2h) with h near 10^-5; for a polynomial with exact fractions you can get the exact slope with no limit and no rounding. The forward quotient (f(a+h)-f(a))/(h) has an error that shrinks in proportion to h; the central one has an error proportional to h^2, because the two first-order errors cancel.

Do not shrink h as far as possible, though. A computer stores each value with a relative rounding error of about 10^-16, and subtracting two nearly equal numbers keeps that absolute error while dividing by a tiny h magnifies it. The run below estimates the derivative of \sin at 1, whose true value is \cos 1=0.5403023058681398: the error falls as h falls, bottoms out (near 10^-8 for the forward quotient and near 10^-5 for the central one) and then climbs again. At h=10^-14 both are wrong in the third digit.

import math

def forward(f, a, h):
    return (f(a + h) - f(a)) / h            # error shrinks like h

def central(f, a, h):
    return (f(a + h) - f(a - h)) / (2 * h)  # error shrinks like h squared

a, true = 1.0, math.cos(1.0)                # the derivative of sin at 1 is cos(1)
for k in (1, 3, 5, 8, 11, 14):
    h = 10.0 ** -k
    print(f"h=1e-{k:<2}  forward {abs(forward(math.sin, a, h) - true):.1e}   central {abs(central(math.sin, a, h) - true):.1e}")

# h=1e-1   forward 4.3e-02   central 9.0e-04
# h=1e-3   forward 4.2e-04   central 9.0e-08
# h=1e-5   forward 4.2e-06   central 1.1e-11
# h=1e-8   forward 3.0e-09   central 2.6e-09
# h=1e-11  forward 1.2e-06   central 1.2e-06
# h=1e-14  forward 3.7e-03   central 3.7e-03

For a polynomial there is no need to estimate at all. Writing f(a+h) as a polynomial in h (a shift of the variable, done by repeated synthetic division) gives f(a)+f'(a) h+…, so the coefficient of h is exactly the slope, as a fraction, with the limit already taken. The code below does that with Fraction and reproduces the hand computation of f'(2)=9 for x^3-3x; the list is the polynomial f(2+h)=2+9h+6h^2+h^3, so the difference quotient is 9+6h+h^2, as before. The same idea is what a computer algebra system such as SymPy does when it differentiates symbolically.

from fractions import Fraction

def shift(coeffs, a):
    """Coefficients of f(a + h) for f = c0 + c1 x + c2 x^2 + ..., by repeated synthetic division."""
    out, cur = [], list(coeffs)
    while cur:
        carry, rest = Fraction(0), []
        for c in reversed(cur):
            carry = c + carry * a
            rest.append(carry)
        out.append(rest.pop())              # the remainder is the next coefficient
        cur = rest[::-1]
    return out

f = [0, -3, 0, 1]                           # x^3 - 3x
print(shift(f, Fraction(2)))                # [Fraction(2, 1), Fraction(9, 1), Fraction(6, 1), Fraction(1, 1)]
print(shift(f, Fraction(2))[1])             # 9, the slope at 2
print(shift([Fraction(4), 0, Fraction(1, 2)], Fraction(6))[1])   # 6, the slope of x^2/2 + 4 at 6

#include <iostream>
#include <cstdio>
#include <cmath>
using namespace std;

// the slope at a of c[0] + c[1] x + c[2] x^2 + ..., exactly, by Horner's rule on the polynomial and its derivative
double slope(const double c[], int n, double a) {
  double p = 0, dp = 0;
  for (int i = n - 1; i >= 0; i--) {
    dp = dp * a + p;
    p = p * a + c[i];
  }
  return dp;
}

int main() {
  double a = 1.0, truth = cos(1.0);                    // the derivative of sin at 1 is cos(1)
  for (int k = 1; k <= 4; k++) {
    double h = pow(10, -k);
    double forward = (sin(a + h) - sin(a)) / h;        // the error shrinks like h
    double central = (sin(a + h) - sin(a - h)) / (2 * h);   // the error shrinks like h squared
    printf("h=1e-%d  forward %.1e   central %.1e\n", k, fabs(forward - truth), fabs(central - truth));
  }
  // for much smaller h, rounding errors take over and both errors grow again

  double f[4] = {0, -3, 0, 1};                         // x^3 - 3x
  printf("%.0f\n", slope(f, 4, 2));                     // 9, the slope at 2
  double g[3] = {4, 0, 0.5};                           // x^2 / 2 + 4
  printf("%.0f\n", slope(g, 3, 6));                     // 6, the slope at 6
  return 0;
}

/* output:
h=1e-1  forward 4.3e-02   central 9.0e-04
h=1e-2  forward 4.2e-03   central 9.0e-06
h=1e-3  forward 4.2e-04   central 9.0e-08
h=1e-4  forward 4.2e-05   central 9.0e-10
9
6
*/

Who invented tangents and derivatives?

No single person did: tangents are ancient, finding them with algebra began in the 1630s, Newton and Leibniz built a calculus on it in the 1660s to 1680s, and the limit definition used today dates from the 1820s.

• Antiquity. Euclid (about 300 BCE) proved the properties of a tangent to a circle in Book III of the Elements, and Apollonius (about 200 BCE) and Archimedes found tangents to conics and to a spiral by special geometric arguments, one curve at a time.
• The 1630s. Pierre de Fermat (about 1636) had a method that compares f(x+e) with f(x), divides by e and then sets e to zero; it found tangents and maximums and is the first form of the difference quotient. René Descartes (La Géométrie, 1637) found the normal to a curve by asking when a circle centered on the axis meets it in a double point, the same double-root idea as in the tangent signature above.
• 1670s and 1680s. Isaac Barrow drew a tiny "differential triangle" on the curve to find tangents. Isaac Newton (notes of 1665 to 1666) treated a changing quantity as a flowing one with a rate, the fluxion, and Gottfried Leibniz (published 1684) wrote the rate as the ratio (dy)/(dx) of tiny differences, a notation still used. They disagreed bitterly about priority. Joseph-Louis Lagrange introduced the prime notation f'(x) in 1797.
• The 1820s. Augustin-Louis Cauchy defined the derivative as the limit of the difference quotient in his lectures at the École Polytechnique (1823), turning a method into a definition. Even then it was believed that continuous functions must be differentiable almost everywhere, and André-Marie Ampère tried to prove it in 1806.
• Counterexamples. Bernard Bolzano (about 1830) and Karl Weierstrass (shown in 1872) built continuous functions that have no derivative at any point, and so no tangent line anywhere, showing how far continuity is from differentiability.

What mistakes do people make with tangent lines and derivatives?

The most common mistakes with tangents and the derivative at a point are the nine below, each with the correct statement.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: tangent lines and the derivative at a point at a glance

• Secant: the line through (a,f(a)) and (a+h,f(a+h)); slope (f(a+h)-f(a))/(h), the average rate of change.
• Derivative at a point: f'(a)=\lim_h→0(f(a+h)-f(a))/(h)=\lim_x→ a(f(x)-f(a))/(x-a), a number: the slope of the tangent and the instantaneous rate.
• Tangent line: y=f(a)+f'(a)(x-a); normal line: slope -(1)/(f'(a)), or the vertical line x=a when the tangent is horizontal.
• Tangent facts: it can meet the curve again and can cross it at an inflection point; for a polynomial, f(x) minus the tangent has a double root at a.
• Meaning: units of y per unit of x; f'(a)>0 rising, <0 falling, =0 horizontal.
• No derivative: a corner, a cusp, a jump, a vertical tangent, or endless wiggling; differentiable implies continuous, and not conversely.
• Code: central difference with h≈10^-5; exact slope of a polynomial from the shift f(a+h).

Glossary: key terms

Tangent line

A straight line that passes through a point of a curve and has the same direction as the curve there; its slope is the derivative of the function at that point.

Secant line

A straight line through two points of a curve; its slope is the average rate of change of the function between the two points.

Difference quotient

The change in a function divided by the change in its input, f of a plus h minus f of a, all over h; it is the slope of a secant line.

Derivative at a point

The limit of the difference quotient as h approaches zero, when it exists; it is the slope of the tangent line and the instantaneous rate of change at that point.

Instantaneous rate of change

How fast a quantity is changing at one exact moment, found as the limit of the average rates of change over shorter and shorter intervals.

Differentiable function

A function that has a derivative at a given point, or at every point of an interval; such a function is continuous there and looks like a straight line when you zoom in.

Normal line

The line through a point of a curve that is perpendicular to the tangent line there; its slope is the negative reciprocal of the derivative when the derivative is not zero.

One-sided derivative

The limit of the difference quotient taken only from the right or only from the left; a function is differentiable at a point exactly when both exist and are equal.

Corner

A point where a continuous graph changes direction abruptly, so the secant slopes from the left and from the right tend to different numbers and there is no tangent line.

Vertical tangent

A tangent line that is the vertical line through the point, which happens when the difference quotient grows without bound; the function has no derivative there, since a vertical line has no slope.

Frequently asked questions

References

1. Joel Hass, Christopher Heil, Maurice D. Weir et al. (2023). Thomas' Calculus: Early Transcendentals in SI Units (15th ed.), Section 3.1: Tangent Lines and the Derivative at a Point. Pearson.
2. James Stewart (2015). Calculus (8th ed.), sections 2.1 The Tangent and Velocity Problems and 2.7 Derivatives and Rates of Change. Cengage Learning.
3. Carl B. Boyer (1949). The History of the Calculus and Its Conceptual Development. Hafner (Dover reprint, 1959).
4. Augustin-Louis Cauchy (1823). Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal. Imprimerie royale.
5. Python Software Foundation. The Python Standard Library: fractions (rational numbers). docs.python.org. https://docs.python.org/3/library/fractions.html.

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