Related Rates

Key Facts

• Start with a relationship among variables, such as $x^2 + y^2 = z^2$ or $V = \frac{4}{3}\pi r^3$.
• Differentiate with respect to time t, not with respect to a single variable unless stated otherwise.
• If $y$ depends on $t$, then $\frac{d}{dt}(y^2) = 2y\frac{dy}{dt}$ by the chain rule.
• For a circle area $A = \pi r^2$, differentiating gives $\frac{dA}{dt} = 2\pi r\frac{dr}{dt}$.
• For a sphere volume $V = \frac{4}{3}\pi r^3$, differentiating gives $\frac{dV}{dt} = 4\pi r^2\frac{dr}{dt}$.
• In ladder and distance problems, a common relation is $x^2 + y^2 = L^2$, so $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$ when $L$ is constant.

Vocabulary

Related rates

A calculus method for finding how one changing quantity varies by using its relationship to other changing quantities.

Chain rule

A differentiation rule used when a variable depends on another variable, such as a quantity depending on time.

Instantaneous rate of change

The rate at which a quantity is changing at one specific moment, usually written as a derivative like dx/dt.

Implicit differentiation

A method of differentiating an equation involving several variables without first solving for one variable explicitly.

Constraint equation

An equation that links the variables in a problem and must remain true as they change.