9.10PureStretch

Connected Rates of Change

When several quantities change with time and are linked by an equation, the chain rule connects their rates. This is the basis of many real-world “how fast is …” problems.

26 min Video by Zeeshan Zamurred Differentiation

Edexcel A level Maths: 9.10 Connected Rates of ChangeWatch the full walkthrough before the notes below.

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What you'll be able to do

Link rates with the chain rule
Set up dV/dt = dV/dr · dr/dt type relations
Identify the known and required rates
Solve connected-rate problems

Connecting rates

If quantities are related, their rates of change are connected by the chain rule. For example, $\frac{d V}{d t} = \frac{d V}{d r} \cdot \frac{d r}{d t}$ .

\frac{d V}{d t} = \frac{d V}{d r} \cdot \frac{d r}{d t}

Chain rule linking rates.

\frac{d V}{d r} = 4 π r^{2}

\frac{d V}{d t} = 4 π r^{2} \cdot \frac{d r}{d t} = 4 π (9) (2)

Answer

72 π

cm³/s

Tip — Write down each rate you know and the one you want, then bridge them with the chain rule.

Formula recap

\frac{d A}{d t} = \frac{d A}{d x} \cdot \frac{d x}{d t}

General connected rates.

V = \frac{4}{3} π r^{3} \Rightarrow \frac{d V}{d r} = 4 π r^{2}

Sphere example.

Common mistakes to avoid

Adding the rates instead of using the chain rule.

Multiply the linking derivatives.

Substituting the value of r before differentiating.

Differentiate first, then substitute.

Key takeaways

Connected rates link via the chain rule.
dV/dt = dV/dr · dr/dt.
List known/required rates, then bridge them; substitute last.

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