The Derivative as a Function

What does it mean for the derivative to be a function?

The derivative of f is the new function f' that gives, for each x, the derivative of f at that x: f'(x)=\lim_h→0(f(x+h)-f(x))/(h), for every x where the limit exists. The derivative at a point is one number at one place. Letting the point move gives a rule that hands out a slope for every input: feed it x and out comes the slope of the graph of f at x. The number f'(a) is simply the value of this function at a.

Four consequences are worth stating at once.

- The domain of f' can be smaller than that of f. It is the set of x in the domain of f at which the limit exists. f(x)=√ x is defined at 0, but f' is not (the graph has a vertical tangent there). - Differentiable has two meanings. f is differentiable at x when f'(x) exists, and differentiable when f' exists at every point of its domain. - There is a second form of the limit. Writing z=x+h, so that h=z-x and h→0 exactly when z→ x, gives f'(x)=\lim_z→ x(f(z)-f(x))/(z-x), which is often easier when f has a root in it, as the examples below show. - Differentiation is an operation on functions. Put a function f in, get a function f' out. That is why the next sections are about finding the formula for f', not a single number.

The slope of the graph of f at x and the instantaneous rate of change of f at x are two readings of the same number f'(x), as in the section on rates of change.

How do you find a derivative function from the definition?

To differentiate from the definition, write the difference quotient with x left as a letter, simplify until h cancels, and then let h→0; what is left is a formula in x. Here are five functions, each with its own trick.

1. f(x)=x^2: expand and cancel. After expanding the bracket, ((x+h)^2-x^2)/(h)=(2xh+h^2)/(h)=2x+h, which tends to 2x.

2. f(x)=x^3. Since (x+h)^3=x^3+3x^2h+3xh^2+h^3, ((x+h)^3-x^3)/(h)=3x^2+3xh+h^2, which tends to 3x^2.

3. f(x)=\frac1x: a common denominator. For x\ne0, using adding fractions, \frac\frac1x+h-\frac1xh=(x-(x+h))/(x(x+h) h)=(-1)/(x(x+h)), which tends to -\frac1x^2.

4. f(x)=(x)/(x-1). Over a common denominator the numerator is (x+h)(x-1)-x(x+h-1)=-h, so the factor h cancels: (f(x+h)-f(x))/(h)=(-1)/((x+h-1)(x-1)), which tends to -(1)/((x-1)^2) for x\ne1.

5. f(x)=√ x: the second form and a conjugate. For x>0, write z-x=(√ z-√ x)(√ z+√ x) and cancel: (√ z-√ x)/(z-x)=(1)/(√ z+√ x), which tends to (1)/(2√ x) as z→ x. This is the same trick as rationalizing. The formula blows up at 0: √ x is defined for x\ge0 and its derivative only for x>0.

Two more cases are worth keeping in mind. For a constant f(x)=c every difference quotient is 0, so f'(x)=0. For a line f(x)=mx+b it is (m(x+h)+b-mx-b)/(h)=m, so f'(x)=m: the slope of a line is its derivative.

Compare x^2→2x with x^3→3x^2. In both the exponent comes down in front and the power of x goes down by one. That pattern is correct, and the differentiation rules prove it for every positive whole number; here it is only an observation from the examples.

To check any of these at a point, put a value of x in: for (x)/(x-1) at x=3 the formula gives -\frac14, and the secant slopes in the tool below close in on exactly that. Try your own function: it expands the difference quotient in powers of h, or reduces the derivative of a quotient, and shows both graphs.

What are the different notations for the derivative?

The derivative of y=f(x) can be written f'(x), y', (dy)/(dx), (df)/(dx), (d)/(dx)f(x) or Df(x); they all name the same function and differ only in how they are written. The prime notation f' and y' goes back to Lagrange and is short. The (d)/(dx) notation is Leibniz's, and it has a use that the prime lacks: it names the variable. If y depends on time t, (dy)/(dt) says what is changing with respect to what, and it carries the units, y-units per t-unit. The letters (d)/(dx) and D stand for the operation of differentiating, which is applied to a function. Read (dy)/(dx) as "the derivative of y with respect to x".

At this stage (dy)/(dx) is a single symbol for a derivative, not a fraction: you cannot cancel the d's, and it is not "dy divided by dx". (Later in calculus dx and dy do get meanings of their own, which is what makes the notation so convenient, but that is a separate idea.)

To give the value of a derivative at a particular number, name the point with a bar and a subscript: f'(a)=(dy)/(dx)\bigg|_x=a=(df)/(dx)\bigg|_x=a. For example, since (d)/(dx)√ x=(1)/(2√ x), (d)/(dx)√ x\bigg|_x=4=(1)/(2·2)=\frac14, which is the slope of the tangent line at (4,2).

Pick whichever form is clearest: f'(x) when you have a named function, (dy)/(dx) when the variables matter, D when you are treating differentiation as an operator.

How do you graph the derivative from the graph of a function?

Read the slope of the graph of f at many points and plot those slopes as heights: where f rises the graph of f' is above the axis, where f falls it is below, and where f has a horizontal tangent it touches the axis. The method is to sketch tangent lines at frequent steps along the graph, estimate each slope, plot the pairs (x,f'(x)) and join them with a curve.

Reading the graph of f' tells you three things at a glance:

1. Where the rate of change is positive, negative or zero. f'>0 means f is rising, f'<0 means falling, f'=0 means a flat spot, which is often the top of a hill or the bottom of a valley.
2. How big the rate is. The farther f' is from the axis, the steeper the graph of f.
3. Where the rate itself is increasing or decreasing. Where the graph of f' climbs, the slope of f is growing; the graph of f bends upward. Where it falls, the graph of f bends downward. The rate of change of f' is the second derivative, written f'', which is just the derivative function applied twice.

Two examples to compare with the tool below. For f(x)=x^2 the slope is negative to the left of the vertex, zero at the vertex and positive to the right, growing steadily: f'(x)=2x, a straight line. For f(x)=x^3-3x the graph rises to a peak at x=-1, falls through x=0 with slope -3 (its steepest fall) and rises again after a valley at x=1, so f'(x)=3x^2-3 is a parabola that crosses the axis at -1 and 1 and has its lowest point at x=0.

The same slope-reading explains why the derivative of \sin x is \cos x: the slope of the sine wave is zero at every peak and valley, largest where the wave crosses the axis going up, and the result is the same wave moved a quarter of a period to the left.

Slide the point below along nine graphs. The top graph shows f and its tangent; the bottom one draws f' up to the point, the height of the dot being the slope of the tangent above it.

What is the derivative of a graph made of line segments?

The derivative of a graph made of line segments is a step function: on each segment it equals the slope of that segment, and at a corner, where two segments with different slopes meet, it has no value. Each segment is a piece of a line y=mx+b, whose derivative is the constant m.

Take the graph through (0,0), (2,4), (5,4), (7,0) and (9,2). Its segments have slopes (4-0)/(2-0)=2, 0, (0-4)/(7-5)=-2 and (2-0)/(9-7)=1, so f' is 2 on (0,2), 0 on (2,5), -2 on (5,7) and 1 on (7,9). At x=2, 5 and 7 the one-sided slopes differ (on the left and right of x=2 they are 2 and 0), so the right-hand and left-hand derivatives disagree and f' does not exist: the graph of f' has an open circle on each side of every corner. At the two ends of the graph only one side exists, a one-sided derivative.

A vertex is only a corner if the slopes really differ. If the pieces on either side lie on the same line, as in (0,0), (1,1), (3,3), (4,1) at x=1, the vertex is not a corner and f' has the value 1 there.

Going backwards works too, with one thing missing. Given the steps of f' you can rebuild the graph by walking along: start at a point, follow slope m_1 to the first break, then m_2, and so on. Starting at (0,0) with slope 3 until x=2, then 0 until 4, then -1 until 6, then 2 until 8 gives the vertices (0,0), (2,6), (4,6), (6,4) and (8,8). The starting point has to be given: moving a graph up or down changes no slope, so a derivative alone determines the graph only up to a constant.

Try both directions below. Type the corners of a graph, or the steps of a derivative, and read off the other one.

Where is a derivative function defined?

A derivative function is defined at the points where the limit exists, which can be fewer than the points where f is defined: it is missing at corners, cusps, vertical tangents and jumps, and at the end of a closed interval only a one-sided derivative can exist. The five ways a derivative can fail are in the section on functions with no derivative; here the question is what that does to the function f'.

Open and closed intervals. f is differentiable on an open interval (finite or infinite) if it has a derivative at every point of it. It is differentiable on a closed interval [a,b] if it is differentiable on (a,b) and the one-sided limits exist at the ends: the right-hand derivative at a and the left-hand derivative at b, \lim_h→0^+(f(a+h)-f(a))/(h), \lim_h→0^-(f(b+h)-f(b))/(h). At an interior point the derivative exists exactly when the left-hand and right-hand derivatives both exist and are equal.

The absolute value. For x>0, |x|=x has derivative 1, and for x<0, |x|=-x has derivative -1. At 0 the right-hand derivative is 1 and the left-hand one is -1, so there is none. The derivative function is -1 for x<0 and 1 for x>0 (for x\ne0 it can be written (x)/(|x|)), with no value at 0. So f=|x| is differentiable on (-∞,0) and on (0,∞) but not at 0, and the domain of f' leaves out the corner.

The square root. At 0 the secant slopes of √ x are (√ h)/(h)=\frac1√ h, which grows without bound, so there is no derivative at 0 and the graph has a vertical tangent there. The domain of f' is (0,∞), one point smaller than that of f.

A derivative need not be continuous. If f is differentiable at a point it is continuous there, but the derivative function can still behave badly. Let f(x)=x^2\sin\frac1x for x\ne0 and f(0)=0. At 0 the difference quotient is h\sin\frac1h, which tends to 0, so f'(0)=0. For x\ne0 the formulas give f'(x)=2x\sin\frac1x-\cos\frac1x, and as x→0 the term 2x\sin\frac1x vanishes but \cos\frac1x keeps swinging between -1 and 1. So f is differentiable everywhere and f' is not continuous at 0. The last graph in the tool above shows it.

How do you compute and plot a derivative function in code?

Differentiate symbolically with SymPy to get an exact formula, or numerically with a central difference on a grid when all you have is numbers; either way the derivative is an object you can evaluate and plot. SymPy works on the formula and returns the formula for f', so diff is differentiation as an operation on a function. It may need simplify to put the answer in the form you would write by hand: for (x)/(x-1) it first returns -(x)/((x-1)^2)+(1)/(x-1), which equals -(1)/((x-1)^2), the result of the definition above.

import sympy as sp

x = sp.symbols('x')
f = x / (x - 1)
print(sp.diff(f, x))              # -x/(x - 1)**2 + 1/(x - 1)
print(sp.simplify(sp.diff(f, x))) # -1/(x - 1)**2
print(sp.diff(sp.sqrt(x), x))     # 1/(2*sqrt(x))
print(sp.diff(1 / x, x))          # -1/x**2

With numbers only, a derivative function is a function that returns a difference quotient. The first block below builds one from any function: derivative(f) returns a new function, so the derivative really is a function in the program too. The second block differentiates samples on a grid with NumPy. On a grid of spacing 0.5 for x^3-3x the interior values are off by exactly h^2=0.25 (the error of a central difference), but at the two ends NumPy can only use one side, a one-sided derivative, and it is off by 2.75: the same endpoint story as on a closed interval.

import numpy as np

def derivative(f, h=1e-5):
    """The derivative of f, as a new function (central difference)."""
    return lambda x: (f(x + h) - f(x - h)) / (2 * h)

g = derivative(lambda t: t**3 - 3*t)
print([round(g(t), 6) for t in (-2, -1, 0, 1, 2)])   # [9.0, 0.0, -3.0, 0.0, 9.0]

xs = np.linspace(-2, 2, 9)                   # spacing 0.5
ys = xs**3 - 3*xs
dy = np.gradient(ys, xs)                     # central inside, one-sided at the ends
print(dy)                                    # [ 6.25  4.    0.25 -2.   -2.75 -2.    0.25  4.    6.25]
print(3*xs**2 - 3)                           # [ 9.    3.75  0.   -2.25 -3.   -2.25  0.    3.75  9.  ]
print(np.round(dy - (3*xs**2 - 3), 4))       # [-2.75  0.25  0.25  0.25  0.25  0.25  0.25  0.25 -2.75]

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

double f(double t) { return t * t * t - 3 * t; }
double derivative(double x, double h) { return (f(x + h) - f(x - h)) / (2 * h); }   // central difference

int main() {
  for (int t = -2; t <= 2; t++) printf("%.4f ", derivative(t, 1e-5));   // 9 0 -3 0 9
  cout << endl;

  double xs[9], ys[9], dy[9];                          // a grid with spacing 0.5
  for (int i = 0; i < 9; i++) { xs[i] = -2 + 0.5 * i; ys[i] = f(xs[i]); }
  dy[0] = (ys[1] - ys[0]) / 0.5;                       // one-sided at the ends
  dy[8] = (ys[8] - ys[7]) / 0.5;
  for (int i = 1; i < 8; i++) dy[i] = (ys[i + 1] - ys[i - 1]) / 1.0;   // central inside
  for (int i = 0; i < 9; i++) printf("%6.2f", dy[i]);
  cout << endl;
  for (int i = 0; i < 9; i++) printf("%6.2f", 3 * xs[i] * xs[i] - 3);   // the exact derivative
  cout << endl;
  for (int i = 0; i < 9; i++) printf("%6.2f", dy[i] - (3 * xs[i] * xs[i] - 3));   // the ends are worst
  cout << endl;
  return 0;
}

/* output:
9.0000 0.0000 -3.0000 0.0000 9.0000
  6.25  4.00  0.25 -2.00 -2.75 -2.00  0.25  4.00  6.25
  9.00  3.75  0.00 -2.25 -3.00 -2.25  0.00  3.75  9.00
 -2.75  0.25  0.25  0.25  0.25  0.25  0.25  0.25 -2.75
*/

Where do the names and symbols of the derivative come from?

The ideas are older than the symbols: the three notations in use today come from Leibniz, Newton and Lagrange, and the word "derivative" itself is French.

• Leibniz. Gottfried Leibniz introduced dx and dy, together with the integral sign, in manuscripts of 1675 and published his differential calculus in 1684. His (dy)/(dx) was meant as a ratio of tiny differences, which is why it is still so convenient when changing variables, and it names the variable.
• Newton. Isaac Newton wrote the rate of change of a flowing quantity y with a dot over the letter, ẏ, which he called its fluxion. The dot survives in physics for derivatives with respect to time.
• Lagrange. Joseph-Louis Lagrange wanted to base calculus on algebra and called the function obtained from f its fonction dérivée, the derived function, in his Théorie des fonctions analytiques (1797). The words "derivative" and the prime f'(x) come from there.
• Arbogast. Louis Arbogast introduced the operator letter D for the operation of differentiating in 1800, which is the notation used for Df(x) and D^2f.

The limit definition that this article starts from is later and is described in the history in the article on tangent lines.

What mistakes do people make with the derivative function?

The most common mistakes with the derivative as a function 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: the derivative as a function at a glance

• Definition: f'(x)=\lim_h→0(f(x+h)-f(x))/(h)=\lim_z→ x(f(z)-f(x))/(z-x), a function of x whose domain is where the limit exists.
• From the definition: x^2→2x, x^3→3x^2, \frac1x→-\frac1x^2, (x)/(x-1)→-\frac1(x-1)^2, √ x→\frac12√ x; constants →0, lines mx+b→ m.
• Notation: f'(x), y', (dy)/(dx), (df)/(dx), (d)/(dx)f(x), Df(x); (dy)/(dx)\big|_x=a for a value; one symbol, not a fraction.
• Graph of f': plot slopes; f'>0 rising, f'<0 falling, f'=0 flat; where f' climbs, f bends upward (f''>0).
• Segments: the derivative is a step function; a corner has no value; the derivative fixes a graph only up to a constant.
• Domain: open interval = every point; closed interval = also one-sided derivatives at the ends; missing at corners, cusps, vertical tangents, jumps.
• Careful: f' need not be continuous (x^2\sin\frac1x); f'(x) is not f(x+h)-f(x) and not \frac1x^2 for \frac1x.
• Code: SymPy diff for the exact formula; a function that returns a central difference, or NumPy gradient, for numbers.

Glossary: key terms

Derivative function

The function f prime whose value at each x is the derivative of f at that x, defined wherever the limit of the difference quotient exists.

Differentiation

The process of finding the derivative of a function; it is an operation that takes a function as input and returns another function.

Differentiable on an interval

Having a derivative at every point of an open interval, or, for a closed interval, on the inside plus one-sided derivatives at the two ends.

Right-hand and left-hand derivatives

The limits of the difference quotient as h approaches zero from above and from below; a derivative exists at an inside point exactly when both exist and are equal.

Prime notation

The way of writing a derivative with a prime, f prime of x or y prime, which goes back to Lagrange.

Leibniz notation

The way of writing a derivative as dy over dx, which names the variable and is read as the derivative of y with respect to x; it is one symbol, not a fraction to cancel.

Step function

A function that is constant on each of several intervals, so its graph is a row of horizontal pieces; it is the derivative of a graph made of line segments.

Second derivative

The derivative of the derivative, written f double prime; it is the rate of change of the slope and tells whether the graph bends upward or downward.

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 3.2: The Derivative as a Function. Pearson.
2. James Stewart (2015). Calculus (8th ed.), section 2.8: The Derivative as a Function. Cengage Learning.
3. Florian Cajori (1929). A History of Mathematical Notations, volume 2, on the notations of the calculus. Open Court Publishing.
4. Joseph-Louis Lagrange (1797). Théorie des fonctions analytiques. Imprimerie de la République.
5. SymPy Development Team. SymPy tutorial: Calculus (diff). docs.sympy.org. https://docs.sympy.org/latest/tutorials/intro-tutorial/calculus.html.
6. NumPy Developers. NumPy reference: numpy.gradient. numpy.org. https://numpy.org/doc/stable/reference/generated/numpy.gradient.html.

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