Derivative as a Rate of Change

Slope at a Point on a Curve

Picture a bicycle moving along a hilly road. Some stretches climb steeply, while others are almost flat. The road's slope changes from point to point. A curved function graph behaves in much the same way.

For a straight line, the slope is the same at every point. For a curved line, the slope changes from point to point. A derivative gives the slope or rate of change at one specific point on a curve.

Gradient of a Secant Line

We begin with a secant line, a straight line through two different points of a curve.

Suppose the curve represents $y = f(x)$ . Choose two distinct points, $P(x, f(x))$ and $Q(x+\Delta x, f(x+\Delta x))$ , on that curve. Here, $\Delta x$ (read "delta x") is the horizontal change from $P$ to $Q$ , so $\Delta x \ne 0$ .

The slope of the secant line through $P$ and $Q$ is:

\begin{aligned} m_{\text{secant}} &= \frac{\Delta y}{\Delta x} \\ &= \frac{f(x + \Delta x) - f(x)}{\Delta x} \end{aligned}

This slope is the average rate of change of $f(x)$ between $P$ and $Q$ .

Secant and Tangent Line Visualization

The secant line connects two distinct points on the curve

y=x^2

, while the tangent line has the same slope as the curve at

P

. In this example, the tangent meets the parabola at

P

. In general, a tangent line may intersect the same curve elsewhere.

From Secant Line to Tangent Line

Next, move $Q$ closer and closer to $P$ . The horizontal distance $\Delta x$ becomes smaller and approaches zero.

As $\Delta x \to 0$ (read "delta x approaches zero"), the secant lines approach the tangent line at $P$ . The tangent line shows the curve's local direction at that point. It need not intersect the entire curve only once. What matters is its local behavior near $P$ .

The tangent's slope is the instantaneous rate of change of the curve at $P$ . We find it with a limit:

\begin{aligned} &m_{\text{tangent}} = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \end{aligned}

Definition of the Derivative

The limit of the secant slopes as $\Delta x$ approaches zero is called the derivative.

The derivative of a function $f(x)$ , denoted as $f'(x)$ (read "f prime x"), is defined as:

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

Finding a derivative is called differentiation.

The derivative $f'(x)$ is a new function. At every value of $x$ where the limit exists, it gives the instantaneous rate of change of $f(x)$ , which is also the slope of the tangent line there. Differential calculus uses the derivative function to analyze both quantities.