Calculus · Quick Reference

Applications of Derivatives Cheat Sheet

Differential Calculus · Lesson 7/20

In one line: derivatives are not just abstract slopes — they describe real rates of change, from the speed of a moving object to how a graph curves.

Key Ideas

1Rates of Change. The derivative of a quantity with respect to time gives its rate of change, such as velocity (rate of change of position).

2Velocity and Acceleration. If s(t) is position, then velocity v(t) = s'(t), and acceleration a(t) = v'(t) = s''(t).

3Increasing/Decreasing Functions. If f'(x) > 0 on an interval, f is increasing there. If f'(x) < 0, f is decreasing.

4Concavity. If f''(x) > 0, the graph is concave up (like a cup). If f''(x) < 0, it is concave down (like a frown).

5Critical Points. Points where f'(x) = 0 or f'(x) is undefined — candidates for local maxima or minima.

Worked Examples

A particle's position is s(t) = t^2 - 4t. Find its velocity at t = 3.

v(3) = 2

For f(x) = x^2 - 6x, find the interval where f is increasing.

f is increasing for x > 3

Find the critical points of f(x) = x^3 - 3x.

Critical points at x = 1 and x = -1

Formulas

v(t) = s'(t)

a(t) = v'(t) = s''(t)

Practice Yourself

If s(t) = 3t^2, find velocity at t=2.

If v(t) = 4t - 1, find acceleration.

For f(x) = x^2 - 4x, find where f is decreasing.

x < 2