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 ${\frac {dy}{dt}}={\frac {2(3)(1)(1)}{3(1)^{2}-(3)^{2}}}=-1$

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