Math 22 Related Rates

Related Variables

We will study problems involving variables that are changing with respect to time. If two or more such variables are related to each other, then their rates of change with respect to time are also related.

Example 1: Given ${\displaystyle y^{3}=x^{2}y+3}$ Find ${\displaystyle {\frac {dy}{dt}}}$ when ${\displaystyle x=3}$, ${\displaystyle y=1}$ and ${\displaystyle {\frac {dx}{dt}}=1}$

Solution: Differentiate both sides of the equation with respect to ${\displaystyle t}$:

${\displaystyle {\frac {d}{dt}}[y^{3}]={\frac {d}{dt}}[x^{2}y+3]}$

${\displaystyle =3y^{2}{\frac {dy}{dt}}=2x{\frac {dx}{dt}}y+x^{2}{\frac {dy}{dt}}}$

${\displaystyle =3y^{2}{\frac {dy}{dt}}-x^{2}{\frac {dy}{dt}}=2xy{\frac {dx}{dt}}}$

${\displaystyle =(3y^{2}-x^{2}){\frac {dy}{dt}}=2xy{\frac {dx}{dt}}}$

Hence, ${\displaystyle {\frac {dy}{dt}}={\frac {2xy{\frac {dx}{dt}}}{3y^{2}-x^{2}}}}$.

Substitute, we get ${\displaystyle {\frac {dy}{dt}}={\frac {2(3)(1)(1)}{3(1)^{2}-(3)^{2}}}=-1}$

Example 2: The revenue ${\displaystyle R}$ from selling ${\displaystyle x}$ units of a product is given by ${\displaystyle R=1200x-x^{2}}$. The sales are increasing at a rate of ${\displaystyle 30}$ units per day. Find the rate of change of the revenue when ${\displaystyle x=400}$

Solution: The sales are increasing at a rate of ${\displaystyle 30}$ units per day. So, ${\displaystyle {\frac {dx}{dt}}=30}$. we want to find the rate of change of the revenue, this means we want to find ${\displaystyle {\frac {dR}{dt}}}$. So, differentiate both sides of the equation with respect to ${\displaystyle t}$ to get:

${\displaystyle {\frac {d}{dt}}[R]={\frac {d}{dt}}[1200x-x^{2}]}$

${\displaystyle ={\frac {dR}{dt}}=1200{\frac {dx}{dt}}-2x{\frac {dx}{dt}}}$

Substitute to get ${\displaystyle {\frac {dR}{dt}}=1200(30)-2(400)(30)=12000}$

Therefore, the rate of change of the revenue is \$12000 per day.

Guidelines for Solving a Related-Rate Problem

 1. Identify all given quantities and all quantities to be determined. If possible, make a sketch
and label the quantities.

2. Identify all given quantities and all quantities to be determined. If possible, make a sketch
and label the quantities.

3. Use the Chain Rule to implicitly differentiate both sides of the equation with respect to
time.

4. After completing Step 3, substitute into the resulting equation all known values for the
variables and their rates of change. Then solve for the required rate of change.