Derivative

The derivative tells us how quickly a function is changing right now.

Two views of one idea

For $y=f(x)$, the derivative $f'(a)$ is both

• the slope of the tangent line to the graph at $x=a$; and
• the instantaneous rate of output change with respect to input at $x=a$.

A graph's slope is a visual way of seeing a rate.

From average to instantaneous

Between $x=a$ and $x=a+h$, the average rate of change is

This is a secant slope. Let the second point approach the first:

First principles: $f(x)=x^2$

At $x=3$, $f'(3)=6$. The tangent slope to $y=x^2$ at $(3,9)$ is $6$.

Notation

“With respect to $x$” identifies which input is changing.

Units

Derivative units are output units divided by input units. If position $s(t)$ is in metres and time in seconds, $s'(t)$ is in metres per second. The second derivative $s''(t)$ is acceleration in metres per second squared.

Efficient differentiation

The definition explains the idea. Rules calculate derivatives efficiently. The power rule is

More complicated functions use the Product Rule, Quotient Rule, and Chain Rule.

Interpreting the sign

• $f'(x)>0$: the function is increasing.
• $f'(x)<0$: the function is decreasing.
• $f'(x)=0$: the tangent is horizontal, but the point is not automatically a maximum or minimum.

A zero derivative identifies a critical-point candidate. The surrounding behaviour decides its type.

When a derivative does not exist

Nearby slopes must approach one finite value. Corners, cusps, vertical tangents, and discontinuities can prevent differentiability.

For example, $f(x)=|x|$ is continuous at zero but not differentiable there: left slopes approach $-1$ and right slopes approach $1$.

Common mistakes

Applications

Derivatives describe velocity, acceleration, marginal cost, growth rate, sensitivity, optimization, and how small input changes affect an output.

Related topics

• Derivative from First Principles
• Derivative as a Rate of Change
• Basic Differentiation Rules
• Tangent Line
• Critical Points
• Optimization