Related Rates Master Reference Cheat Sheet

Key Facts

• In related rates, every changing variable is treated as a function of time, so $\frac{d}{dt}[x^2] = 2x\frac{dx}{dt}$.
• The standard process is identify variables, write a relation, differentiate with respect to $t$, substitute known values, and solve for the unknown rate.
• For a right triangle, $x^2 + y^2 = z^2$ gives $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 2z\frac{dz}{dt}$.
• For a circle, $A = \pi r^2$ gives $\frac{dA}{dt} = 2\pi r\frac{dr}{dt}$.
• For a sphere, $V = \frac{4}{3}\pi r^3$ gives $\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}$.
• For a cone, $V = \frac{1}{3}\pi r^2h$, and similar triangles often create a relation such as $r = kh$ before differentiating.
• A positive rate means the quantity is increasing, and a negative rate means the quantity is decreasing, such as $\frac{dx}{dt} < 0$ for distance getting smaller.
• In shadow problems, similar triangles give proportional equations such as $\frac{H}{L} = \frac{h}{s}$ before differentiating.

Vocabulary

Related rates

A calculus method for finding how fast one quantity changes by using its relationship to another changing quantity.

Implicit differentiation

Differentiating an equation without first solving for one variable, while using the chain rule for variables that depend on $t$.

Rate of change

A derivative such as $\frac{dx}{dt}$ that measures how a quantity changes with respect to time.

Constraint equation

An equation that connects the variables in a problem, such as $x^2 + y^2 = z^2$ or $V = \pi r^2h$.

Similar triangles

Triangles with equal angle measures and proportional side lengths, often used to relate heights, distances, and shadows.

Instantaneous rate

The rate of change at one exact moment, found after differentiating and then substituting the given values.