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

Solution: Differentiate both sides of the equation with respect to $t$:

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

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

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

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

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

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

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.

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