Differentiation (Rates of Change)

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Rates Chain: dy/dx × dx/dt = dy/dt

\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}

Every rates question: given one rate, find another — the middle derivative comes from a formula linking the variables.

Reflex: Write the chain rule FIRST.

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Differentiation (Rates of Change)

4 worked examples

Rates Chain: dy/dx × dx/dt = dy/dt

\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}

Every rates question: given one rate, find another — the middle derivative comes from a formula linking the variables.

Reflex: Write the chain rule FIRST.

Rates: The Five-Step Chain Rule Routine

\frac{d\square}{dt}

Write given & wanted rates → chain rule link → middle derivative from an equation (area/volume/similar triangles/curve) → substitute → evaluate at the instant (solve for $x$ , $r$ , or $h$ if needed).

Reflex: Summarize as: know $\frac{dA}{dt}$ , want $\frac{dB}{dt}$ , need $B$ -vs- $A$ equation.

Cones: Similar Triangles Before Differentiating

\frac{r}{9} = \frac{h}{24} \Rightarrow r = \frac{3h}{8} \;\Rightarrow\; V = \frac{3}{64}\pi h^3

Two variables in $V=\frac{1}{3}\pi r^2h$ ? Tie $r$ to $h$ via similar triangles, substitute, THEN differentiate.

Reflex: Two variables in the formula? Eliminate one geometrically before touching $\frac{d}{dh}$ .

Chain Rule: Relative Speeds Multiply

\frac{dc}{da} = \frac{dc}{db}\times\frac{db}{da}

Charlie's rate vs Andy = Charlie vs Bobby × Bobby vs Andy — rates chain through the middleman.

Reflex: Rates chain by multiplication, exactly like relative speeds.

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