How to Interpret a Derivative in Context

In How to Interpret a Derivative in Context, separate the amount from its rate and the rate from its rate of change. Position, velocity, and acceleration may be evaluated at the same time, but they answer different questions and carry different units. A negative value describes direction or signed change; it does not automatically mean the quantity is small or slowing.

When reading a context, write a sentence for each derivative before calculating. Include what changes, with respect to what, at which input, and in what units. Then use signs to describe direction and compare signs of velocity and acceleration to determine whether speed is increasing or decreasing.

A derivative value is a compact sentence. Its units say what is changing per unit of what. Its sign says whether the output rises or falls as the input increases. Its magnitude says how sensitive the output is near that state.

For a small input change $\Delta x$, the derivative also predicts $\Delta f\approx f'(x)\Delta x$. That approximation gives the derivative operational meaning: it estimates what a nearby real change will do.