What exactly does a derivative measure?

The rate at which a function changes at a given point — how much the value moves per unit of the argument. Geometrically it is the slope of the tangent to the graph there. If the argument is time and the value is distance, the derivative is the instantaneous speed.

How is the difference quotient different from the derivative?

The difference quotient is the slope of a secant, the average rate of change across the whole stretch from x₀ to x₀ + h. The derivative comes out of it by the limit h → 0: the secant rotates about the point and becomes a tangent. So the quotient describes an interval and the derivative a single point.

Why is the derivative of a product not the product of the derivatives?

Because when the argument moves, both factors move at once, and the change in the product is made of two contributions: the first factor changing while the second is held, and the other way round. Hence f′g + fg′. The shortest counterexample: with f = g = x we get (x · x)′ = 2x, while the product of the derivatives is 1 · 1 = 1.

Does every function have a derivative?

No. A derivative exists where the limit of the difference quotient exists and is finite, so the function has to be continuous there and free of corners. The function |x| is continuous at zero, but its slope is −1 on the left and +1 on the right, so it has no derivative at zero.

What does a negative derivative at a point mean?

That the function is decreasing there: as the argument grows the value drops and the tangent leans downwards. A positive derivative means growth, and a derivative of zero means a horizontal tangent — a candidate for an extremum. That observation is the subject of the next lesson.

How do I differentiate a square root and the fraction 1/x?

By writing them as powers and using the same rule: √x = x^(1/2) gives ½ · x^(−1/2), that is 1/(2√x), and 1/x = x^(−1) gives −1 · x^(−2), that is −1/x². The power rule holds for any exponent, not only for the natural ones.

What notation is used for a derivative?

Most often f′(x) (Lagrange notation) or dy/dx (Leibniz notation) — the two mean the same thing. The result itself can be written in many equivalent ways: 2x + 3, 3 + 2x and 2(x + 1) + 1 are one and the same function. The exercises accept any correct spelling, because what is checked is equality of functions, not of strings.

Advanced level

The derivative

Q: How do I spot that the chain rule is needed?

By the function being a composition: something substituted into something else, such as (2x + 1)⁵, √(3x − 4) or sin(4x). Differentiate the outer function as if its inside were a plain x, then multiply by the derivative of the inside. Forgetting that multiplication is the single most common mistake in the whole lesson.

A derivative measures how fast a function changes at a point — it is the slope of the tangent to the graph. Meet the definition through the difference quotient and the six rules that compute it without taking a limit every time.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Limit of a functionA limit says where the values of a function are heading as the argument closes in on a number — even when the function has no value at that number at all. It is the first idea of analysis and the ground the derivative stands on.
PowersA power is shorthand for multiplying the same factor by itself. Learn the base and the exponent, the laws of exponents, zero and negative exponents, and scientific notation.

All formulas

Definition of the derivative
$f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$
the limit of the difference quotient as the increment shrinks to zero
Power rule
$(x^n)' = n \cdot x^{n-1}$
the exponent steps in front of x and drops by one
Constants and sums
$(c \cdot f)' = c \cdot f', \qquad (f + g)' = f' + g'$
a constant factor passes through; a sum differentiates term by term
Product rule
$(f \cdot g)' = f' g + f g'$
the derivative of a product is NOT the product of the derivatives
Quotient rule
$\left(\frac{f}{g}\right)' = \frac{f' g - f g'}{g^2}$
a difference on top — the order matters
Chain rule
$\big(f(g(x))\big)' = f'(g(x)) \cdot g'(x)$
the outer derivative times the derivative of the inside
Root and reciprocal
$(\sqrt{x})' = \frac{1}{2\sqrt{x}}, \qquad \left(\frac{1}{x}\right)' = -\frac{1}{x^2}$
the same power rule with exponents ½ and −1
Trigonometric functions
$(\sin x)' = \cos x, \qquad (\cos x)' = -\sin x$
the cosine picks up a minus sign

The graph of a function is rarely a straight line, so the question "how fast does it grow" has no single answer — at every point it grows differently. The derivative is the tool that gives that rate for one point.

The difference quotient: an average rate of change

Start with something we can already compute: the slope of a secant, the line through two points of the graph.

A secant through A(1, 1) and B(3, 9). From 1 to 3 the value rose by 8 while the argument rose by 2, so the average rate of change is 4.

For $f(x) = x^2$ , the point $x_0 = 1$ and an increment $h = 2$ :

\frac{f(1 + h) - f(1)}{h} = \frac{9 - 1}{2} = 4

This fraction is the difference quotient. It only speaks about the whole stretch, though — and what we want is the rate exactly at $1$ .

From secant to tangent

So let $h$ shrink. The point $B$ slides down the curve towards $A$ and the secant rotates about $A$ :

$h$	point $B$	difference quotient
$2$	$(3, 9)$	$4$
$1$	$(2, 4)$	$3$
$0.5$	$(1.5,\ 2.25)$	$2.5$
$0.1$	$(1.1,\ 1.21)$	$2.1$
$0.01$	$(1.01,\ 1.0201)$	$2.01$

The numbers head for $2$ — and that is no coincidence, because the difference quotient simplifies:

\frac{(1 + h)^2 - 1^2}{h} = \frac{1 + 2h + h^2 - 1}{h} = \frac{h(2 + h)}{h} = 2 + h

As $h \to 0$ only $2$ remains. The secant stops being a secant: it touches the graph at a single point and becomes the tangent.

The limit of those secants: the tangent to the parabola at (1, 1). Its slope is 2 — exactly the value of the derivative there.

The definition

f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}

The derivative of $f$ at $x_0$ is the limit of the difference quotient as the increment tends to zero. Note that at $h = 0$ the fraction reads $\tfrac{0}{0}$ — without the limit from the previous lesson this definition would make no sense at all.

Three readings of the same number:

geometric — the slope of the tangent to the graph at $x_0$ ;
physical — the instantaneous speed, when the argument is time and the value is distance;
practical — roughly how much the value moves when the argument grows by one unit.

Asking for the derivative at every point at once produces a new function, written $f'$ — and that function is usually what the computation is after.

Use the definition to differentiate f(x) = x² + 3x at an arbitrary point x.

The power rule

Computing every derivative from the definition would be punishing. Fortunately the definition yields rules that are then applied mechanically. The most important one:

(x^n)' = n \cdot x^{n-1}

The exponent steps in front of $x$ and drops by one. The rule holds for any exponent — fractional and negative included, which settles roots and reciprocals:

function	as a power	derivative
$c$ (a constant)	$c \cdot x^0$	$0$
$x$	$x^1$	$1$
$x^3$	$x^3$	$3x^2$
$\sqrt{x}$	$x^{1/2}$	$\dfrac{1}{2\sqrt{x}}$
$\dfrac{1}{x}$	$x^{-1}$	$-\dfrac{1}{x^2}$
$\dfrac{1}{x^2}$	$x^{-2}$	$-\dfrac{2}{x^3}$
$\sin x$	—	$\cos x$
$\cos x$	—	$-\sin x$

The derivative of a constant is zero, because a constant function does not change — its graph is horizontal, and the slope of a horizontal line is $0$ .

Constants, sums and polynomials

Two rules that break any polynomial into pieces:

(c \cdot f)' = c \cdot f', \qquad (f + g)' = f' + g'

A constant factor passes through the derivative untouched, and a sum is differentiated term by term. Together with the power rule that is a recipe for any polynomial:

(4x^3 - 5x^2 + 7x - 9)' = 12x^2 - 10x + 7

Every term on its own, and the constant term disappears.

The product rule

(f \cdot g)' = f' g + f g'

The derivative of a product is not the product of the derivatives — worth checking on the simplest example there is. With $f = g = x$ we get $(x \cdot x)' = (x^2)' = 2x$ , while the product of the derivatives is $1 \cdot 1 = 1$ .

Differentiate f(x) = (2x + 1)(x² − 3).

The quotient rule

\left(\frac{f}{g}\right)' = \frac{f' g - f g'}{g^2}

There is a difference on top, so the order matters — swapping the two terms flips the sign of the whole answer. The denominator is squared.

Differentiate f(x) = (3x + 2)/(x − 4).

The chain rule

\big(f(g(x))\big)' = f'(g(x)) \cdot g'(x)

When a function is a composition — something substituted into something else — differentiate the outer function as if its inside were a plain $x$ , then multiply by the derivative of the inside. That multiplication is the easiest thing in calculus to forget.

\big((2x + 1)^5\big)' = 5(2x + 1)^4 \cdot 2 = 10(2x + 1)^4

\big(\sin 3x\big)' = \cos 3x \cdot 3 = 3\cos 3x

\big(\sqrt{4x + 5}\big)' = \frac{1}{2\sqrt{4x + 5}} \cdot 4 = \frac{2}{\sqrt{4x + 5}}

Differentiate f(x) = (x² − 3)⁴ and point out where the derivative of the inside shows up in the answer.

Exercises

The exercises walk through every rule in this lesson in turn: a power with a non-integer exponent, a polynomial, a product, a quotient, a composition, and the value of a derivative at a point. Where the question asks for $f'(x)$ the answer is a function — type it in any correct form, because what is graded is equality of functions rather than of strings (6x^2 + 2x - 6 and 6x²+2x−6 pass alike). Only the last kind of question, about $f'(x_0)$ , expects a number.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

f(x) = 2/x → f′(x)

Common mistakes

Dropping the derivative of the inside — $\big(\sin 3x\big)' = 3\cos 3x$ , not $\cos 3x$ .
Multiplying derivatives instead of using the product rule — $(fg)' = f'g + fg'$ , never $f'g'$ .
Reversing the quotient rule — the numerator is $f'g$ first, then $fg'$ subtracted.
A constant differentiating to a constant — a constant function does not change, so its derivative is $0$ .
Losing the minus in 1/x and in the cosine — $\left(\tfrac{1}{x}\right)' = -\tfrac{1}{x^2}$ and $(\cos x)' = -\sin x$ .
Confusing f′(x) with f′(x₀) — the first is a function, the second a number: that function's value at one point.

Formula card

Topic: The derivative

Definition of the derivative
$f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}$
the limit of the difference quotient as the increment shrinks to zero
Power rule
$(x^n)' = n \cdot x^{n-1}$
the exponent steps in front of x and drops by one
Constants and sums
$(c \cdot f)' = c \cdot f', \qquad (f + g)' = f' + g'$
a constant factor passes through; a sum differentiates term by term
Product rule
$(f \cdot g)' = f' g + f g'$
the derivative of a product is NOT the product of the derivatives
Quotient rule
$\left(\frac{f}{g}\right)' = \frac{f' g - f g'}{g^2}$
a difference on top — the order matters
Chain rule
$\big(f(g(x))\big)' = f'(g(x)) \cdot g'(x)$
the outer derivative times the derivative of the inside
Root and reciprocal
$(\sqrt{x})' = \frac{1}{2\sqrt{x}}, \qquad \left(\frac{1}{x}\right)' = -\frac{1}{x^2}$
the same power rule with exponents ½ and −1
Trigonometric functions
$(\sin x)' = \cos x, \qquad (\cos x)' = -\sin x$
the cosine picks up a minus sign

A secant through A(1, 1) and B(3, 9) on the parabola y = x². Its slope of 4 is the difference quotient — the average rate of change from 1 to 3.

The tangent to y = x² at the point (1, 1). Its slope of 2 is the value of the derivative: f′(1) = 2.

Frequently asked questions

A luminous parabola and a straight line crossing above a faint coordinate grid, the vertex flaring like a star, against a cosmic nebula in violet and magenta

Functions and analysis

Why a function's graph can never double back

The whole definition of a function comes down to one word: uniqueness. One argument, one value — and not a single one more. That seemingly minor condition decides why a circle is not the graph of a function, why a line and a parabola describe so much of the world, and why four datasets with identical summary statistics can look nothing alike.

12 min