What Is a Derivative? A Plain Explanation

The idea before the notation

The question 'what is a derivative' has a short answer: it is the rate at which a function is changing at one particular point. Not the average rate over an interval — the rate at a single instant.

Speed is the familiar case. Driving 120 miles in two hours gives an average of 60 miles per hour, but the speedometer does not show an average; it shows what the rate is right now. The derivative is the mathematical version of that reading.

Why it equals the slope of a tangent

On a graph, the average rate of change between two points is the slope of the straight line joining them. Bring the second point closer to the first and that line pivots. Keep going, and it settles onto the line that just touches the curve at that point — the tangent.

The derivative is the slope of that tangent. This is why a derivative of zero marks a peak or a trough: a horizontal tangent means the function is momentarily neither rising nor falling.

Where the power rule comes from

The rules are shortcuts for a limit you would otherwise compute every time. For xⁿ, expanding the difference quotient and letting the gap shrink leaves exactly one surviving term: nxⁿ⁻¹. Every other term contains a factor of the gap and vanishes.

The constant vanishing is not an accident of the rule. A constant never changes, so its rate of change is zero — which also explains why an indefinite integral needs '+ C': the original constant is unrecoverable.

The rules you will actually use

Most of a first calculus course is these five, applied in combination.

The minus sign on cos x is the one people lose, and losing it flips the answer entirely. eˣ being its own derivative is not a coincidence either — e is defined as the base where the growth rate equals the current value.

Reading what a derivative tells you

The sign and size of f′(x) describe the original function's behaviour at that point:

• Positive: the function is increasing there.
• Negative: it is decreasing.
• Zero: it is momentarily flat — a maximum, a minimum, or a point of inflection.
• Large in size: it is changing quickly, in whichever direction the sign indicates.

Differentiating a second time gives the rate at which the rate is changing, which distinguishes a peak from a trough and describes how the curve bends.

Notation, and why there is more than one

Two notations appear constantly and mean the same thing. f′(x), read 'f prime of x', is compact and suits a function that already has a name. dy/dx names both variables and is clearer whenever it matters which quantity is changing with respect to which — in physics, where a quantity might be differentiated with respect to time or position, that distinction is doing real work.

A second derivative is written f″(x) or d²y/dx². It is simply the derivative of the derivative: the rate at which the rate itself is changing. On a graph it describes how the curve bends, which is what separates a maximum from a minimum at a stationary point.

When the simple rules are not enough

Term-by-term differentiation covers sums and constant multiples. Two situations need more:

• A product of two functions, like x²·sin x, needs the product rule.
• A function inside another, like sin(2x) or (x + 1)³, needs the chain rule.

Applying the power rule to sin(2x) as if it were a simple term is the most common error in early calculus. If there is anything other than x inside a function, the chain rule applies.

Questions about what is a derivative