A derivative measures how fast a function’s output changes at one exact input: its instantaneous rate of change. On a graph it is the slope of the tangent line at that point. The derivative of f(x) = x2 is f′(x) = 2x, so at x = 3 the curve is rising with slope 6.
In Spanish
derivada
Also called
f prime, dy/dx, instantaneous rate of change, slope of the tangent line, differentiation
Grade level
Calculus
CalculusAP
Derivative in a picture
Slide the second point toward (1, 1) and the secant slopes close in on 2. That limit is the derivative: f′(1) = 2.
Formal definition and notation
The derivative of f at a is f′(a) = limh → 0f(a + h) − f(a)h, provided this limit exists. It is the limit of the slopes of secant lines through (a, f(a)) as the second point slides toward the first. The function f′(x) that gives this value at every x is called the derivative function.
Symbol
Read it as
Meaning
f′(x)
f prime of x
Lagrange’s notation for the derivative of f
dydx
d y d x
Leibniz’s notation: the rate of change of y with respect to x
f′′(x)
f double prime of x
the second derivative, the rate of change of the rate of change
Key facts
Meaning: the derivative is the slope of the curve at a point, or the rate of change at an instant: velocity is the derivative of position.
f′(x) > 0: the function is increasing there. f′(x) < 0: decreasing. f′(x) = 0: a flat tangent, often a peak or a valley.
The derivative of a line y = mx + b is just m: a line has the same slope everywhere.
Units are output units per input unit, such as meters per second or dollars per item.
A derivative may not exist at a corner (like |x| at 0), a jump, or a vertical tangent.
Limit definition and the power rule
Limit definition and the power rule
f′(x) = limh → 0f(x + h) − f(x)h
h
a small change in x that shrinks to 0
f(x + h) − f(x)h
the slope of a secant line (rise over run)
ddxxn = nxn − 1
the power rule, the shortcut you use most
Constants have derivative 0, and you can differentiate a sum term by term: ddx(3x2 − 4x + 7) = 6x − 4.
A straight line has one slope everywhere. A curve has a different slope at each point, and the derivative gives the slope at any point you choose.
Derivativevs.Average rate of change
The average rate is the slope of a secant line between two points; the derivative is the slope at one point. For x2 from 1 to 2 the average rate is 3, but f′(1) = 2.
An integral adds up a quantity, like area or total distance; a derivative finds a rate. They are inverse operations, linked by the Fundamental Theorem of Calculus.
Quick check: 3 questions
Question 1 of 3
What is the derivative of f(x) = 7x3?
Show answer and explanation
Answer: A. Power rule: 7 × 3x2 = 21x2. 21x3 forgets to lower the exponent, 7x2 drops the 3, and 74x4 is the antiderivative.
Question 2 of 3
If f′(5) = 0, what does the graph of f look like at x = 5?
Show answer and explanation
Answer: D. A derivative of 0 means the tangent line has slope 0: it is horizontal, often at a peak or valley. Crossing the x-axis would mean f(5) = 0, not f′(5) = 0.
Question 3 of 3
A company’s cost is C(x) = 0.5x2 + 10x dollars for x items. What is C′(20)?
Show answer and explanation
Answer: B.C′(x) = x + 10, so C′(20) = 30 dollars per item: the approximate cost of one more item. 400 is C(20), the total cost.
Derivative: frequently asked questions
What does a derivative tell you?
How fast something is changing at one instant. On a graph it is the slope of the tangent line, so a positive derivative means the function is rising and a negative one means it is falling. If s(t) is position, s′(t) is velocity; if C(x) is cost, C′(x) is about the cost of one more item.
What is the power rule?
For any numbern, the derivative of xn is nxn − 1: bring the exponent down in front and subtract 1 from it. So x3 becomes 3x2, x becomes 1, and x−1 becomes −x−2. Combined with the rules for constants and sums, it handles every polynomial.
What is the difference between dy/dx and f prime?
Nothing in meaning; they are two notations for the same derivative. f′(x), from Lagrange, is handy when the function has a name. dydx, from Leibniz, shows which variable changes with respect to which, and it is common in science and in the chain rule. Both mean the instantaneous rate of change.
LimitA limit is the value a function gets closer and closer to as the input approaches a certain number. We write limx → af(x) = L. The function need not even be defined at a. For f(x) = x2 − 1x − 1, f(1) is undefined, but the limit as x → 1 is 2.Calculus · Precalculus, CalculusSlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1SecantSecant (sec) is the trigonometric function equal to 1 divided by cosine: sec θ = 1cos θ. In a right triangle it is the hypotenuse divided by the side adjacent to the angle. Secant is undefined wherever cosine is 0, such as at 90°, and its values are always at least 1 or at most −1.Trigonometry · Algebra 2, Precalculus, CalculusRate of ChangeA rate of change tells how much an output changes for each 1-unit change in the input: the change in y divided by the change in x. If a phone plan costs 15 dollars more for every extra hour, its rate of change is 15 dollars per hour. For a straight line, the rate of change equals the slope.Functions · Grade 8, Algebra 1, Algebra 2MeterA meter (m) is a metric unit of length equal to 100 centimeters. It is the base unit of length in the metric system, a little over three feet (about 3.28 feet). A doorway is about 2 meters tall, and 1,000 meters make a kilometer. To change meters to centimeters, multiply by 100.Measurement · Grade 3, Grade 4, Grade 5Power RuleThe power rule is the shortcut for the derivative of a power of x: if f(x) = xn, then f′(x) = nxn − 1. Bring the exponent down as a coefficient and subtract 1 from it, so x5 becomes 5x4. It works for any real exponent, including negative and fractional ones.Calculus · CalculusExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8SubstituteSubstitute means to replace a variable with a number or with an equal expression. In 4x − 1, substituting 3 for x gives 4(3) − 1. You substitute to evaluate expressions, to check whether a number is a solution, and to solve systems of equations by swapping in an equal expression.Algebra · Grade 6, Grade 7, Grade 8HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7IntegralAn integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral ∫baf(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: ∫ 2xdx = x2 + C.Calculus · CalculusInverse OperationsInverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.Operations · Grade 1, Grade 3, Grade 6NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2FunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2Chain RuleThe chain rule is the rule for the derivative of a composite function: if y = f(g(x)), then y′ = f′(g(x)) · g′(x). Differentiate the outer function, leave the inner function inside, then multiply by the derivative of the inner function. For (x2 + 1)3, the derivative is 3(x2 + 1)2 · 2x.Calculus · CalculusCalculusCalculus is the branch of mathematics that studies how quantities change. Its two main tools are the derivative, which measures the rate of change at an instant (the slope of a curve at a point), and the integral, which adds up infinitely many tiny pieces (the area under a curve). Both are built on limits.Calculus · CalculusPolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6