a Derivative?

The instantaneous rate of change of a function — the slope of the curve at a single, exact point.

Simple Definition

Imagine you're driving a car. Your speedometer doesn't show your average speed over the whole trip — it shows how fast you're going right now, at this exact moment. That's a derivative.

In mathematics, a derivative tells you how fast a function is changing at any given point. If you draw the graph of a function, the derivative at a point gives you the slope of the tangent line — the line that just barely touches the curve at that point.

A positive derivative means the function is going up. A negative derivative means it's going down. A derivative of zero means the function has leveled off — you're at a peak, a valley, or a flat section.

Intuition

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Think of it this way

Position is a function of time: f(t). The derivative f′(t) is your velocity — how quickly your position is changing. The derivative of velocity is acceleration — how quickly your velocity is changing. Every time you ask "how fast is this changing?", you're asking for a derivative.

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The geometric picture

Take two points on a curve and draw a line through them — that's a secant line, and its slope is the average rate of change. Now slide the two points together until they become one. The secant line becomes a tangent line, and its slope is the derivative. The derivative is what you get when the gap between the two points shrinks to zero.

Visual

Drag the point along the curve to see the tangent line and derivative value change

Formal Definition

The derivative of a function $f$ at a point $x$ is defined as the limit:

Definition via limit

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

This fraction $\dfrac{f(x+h)-f(x)}{h}$ is the slope of the secant line between two points separated by a distance $h$. As $h \to 0$, the secant line becomes the tangent line, and we get the instantaneous rate of change.

The derivative is also written as $\dfrac{dy}{dx}$ (Leibniz notation) or $\dot{f}$ (Newton's dot notation).

Key Formulas

Constant	$\dfrac{d}{dx}[c] = 0$
Power Rule	$\dfrac{d}{dx}[x^n] = nx^{n-1}$
Exponential	$\dfrac{d}{dx}[e^x] = e^x$
Natural Log	$\dfrac{d}{dx}[\ln x] = \dfrac{1}{x}$
Sine	$\dfrac{d}{dx}[\sin x] = \cos x$
Cosine	$\dfrac{d}{dx}[\cos x] = -\sin x$
Sum Rule	$\dfrac{d}{dx}[f+g] = f' + g'$
Product Rule	$\dfrac{d}{dx}[fg] = f'g + fg'$
Chain Rule	$\dfrac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$

Quick Examples

Example 1 — Power Rule

$f(x) = x^3$

Apply $\frac{d}{dx}[x^n] = nx^{n-1}$:
$f'(x) = 3x^2$

Example 2 — Sum Rule

$f(x) = x^2 + 5x + 3$

Differentiate term by term:
$f'(x) = 2x + 5$

Example 3 — Chain Rule

$f(x) = \sin(x^2)$

Outer: $\cos$, inner: $x^2$:
$f'(x) = 2x\cos(x^2)$

Example 4 — Product Rule

$f(x) = x^2 e^x$

$f'g + fg'$:
$f'(x) = 2xe^x + x^2e^x$

Common Mistakes

✗ Forgetting the Chain Rule

When differentiating a composed function like $\sin(x^2)$, you must multiply by the derivative of the inner function. Writing $\cos(x^2)$ without the $2x$ is the most common error in Calculus I.

✗ Confusing the derivative of a product

$\frac{d}{dx}[f \cdot g] \neq f' \cdot g'$. You must use the product rule: $f'g + fg'$. Multiplying the derivatives is wrong.

✗ Treating the derivative of e^x incorrectly

$\frac{d}{dx}[e^x] = e^x$, not $xe^{x-1}$. The power rule does not apply to exponential functions — that rule is for $x^n$ where $n$ is a constant, not the exponent.

Related Concepts

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Constant	\(\dfrac{d}{dx}[c] = 0\)
Power Rule	\(\dfrac{d}{dx}[x^n] = nx^{n-1}\)
Exponential	\(\dfrac{d}{dx}[e^x] = e^x\)
Natural Log	\(\dfrac{d}{dx}[\ln x] = \dfrac{1}{x}\)
Sine	\(\dfrac{d}{dx}[\sin x] = \cos x\)
Cosine	\(\dfrac{d}{dx}[\cos x] = -\sin x\)
Sum Rule	\(\dfrac{d}{dx}[f+g] = f' + g'\)
Product Rule	\(\dfrac{d}{dx}[fg] = f'g + fg'\)
Chain Rule	\(\dfrac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)