What Is a Derivative? The Difference Quotient and the Slope of a Curve

Q: What is the difference quotient?

f(x + h) - f(x)/h: the slope of the line through two points on a curve a distance h apart. Its limit as h → 0 is the derivative.

How the slope between two points becomes the slope at one point, what the difference quotient is, a worked derivative of $x^2$ from first principles, and why every earlier topic in Foundations feeds into this idea.

3 min read Supports 3 lessons

The slope formula needs two points. A curve's steepness changes from point to point, so how can we talk about its slope at one point? The answer is the central idea of calculus: take two points, then push them together.

The difference quotient

Keep one point at $x$ and put the other a small step $h$ further along, at $x + h$ . The slope of the secant between them is called the difference quotient.

Difference quotient

\frac{f(x + h) - f(x)}{h}

Change in output over change in input, with the run written as $h$ .

Now shrink $h$ . Drag the slider, or press the button to watch it happen:

As $h$ heads to $0$ , the secant line turns into the tangent line, which just touches the curve, and its slope settles on a single number: $2$ .

The derivative of x² from first principles

$f(x) = x^2$
Work out the difference quotient in general.

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Derivative

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

Key idea

This is the definition every calculus course starts from. It turns "slope at a point" from a vague idea into a calculation.

What it means

The derivative is a rate of change at an instant. If $f(t)$ is position, $f'(t)$ is velocity. If $f$ is the cost of making $x$ items, $f'$ is the cost of one more. Where $f'(x) = 0$ , the tangent is flat, which is where a graph turns: a maximum or a minimum.

Everything comes together

Look at what that short calculation used: function notation, substitution, expanding brackets, simplifying fractions, slope, and limits. The binomial theorem extends it to $x^n$ , giving $nx^{n-1}$ . Trig identities, radians and $e$ are chosen precisely so that derivatives of sine, cosine and $e^x$ come out cleanly.

Worked example

Use the difference quotient to find the slope of $f(x) = x^2 + 3x$ at $x = 2$ .

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Simplify the difference quotient.
$\frac{(x + h)^2 + 3(x + h) - x^2 - 3x}{h} = 2x + 3 + h$

Watch out

You cannot set $h = 0$ straight away: that gives $\frac00$ . Simplify first so the $h$ in the denominator cancels, then let $h$ go to zero.

Tip

You are ready for calculus. Everything in Foundations has been building towards this one limit.

Common questions

What is a derivative in simple terms?: It is the slope of a curve at a single point, or equivalently how fast a quantity is changing at an instant.
What is the difference quotient?: $\frac{f(x + h) - f(x)}{h}$ : the slope of the line through two points on a curve a distance $h$ apart. Its limit as $h \to 0$ is the derivative.

Practise the derivative

The The Difference Quotient lesson has worked examples and unlimited practice that adapts to you.

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