Related rates

A balloon's radius grows, so its volume grows. Their rates are linked because the quantities are. Differentiating the formula with respect to time links them.

A 10 ft ladder against a wall:

x^{2} + y^{2} = 100

. Pull the base out, and the top slides down.

Draw it and name the quantities.
Write an equation that links them.
Differentiate with respect to $t$ .
Substitute what you know, then solve.

Work through it

A 10 ft ladder's base slides out at 1 ft/s. How fast does the top fall when $x = 6$ ?

Link them. $x^{2} + y^{2} = 100$

Differentiate in $t$ . $2 x \frac{d x}{d t} + 2 y \frac{d y}{d t} = 0$

When $x = 6$ , $y = 8$ . $12 (1) + 16 \frac{d y}{d t} = 0$

Solve. $\frac{d y}{d t} = - \frac{3}{4}$ ft/s

A circle's radius grows at 2 cm/s. How fast is its area growing when $r = 5$ ?

Link them. $A = π r^{2}$

Differentiate. $\frac{d A}{d t} = 2 π r \frac{d r}{d t}$

Substitute. $20 π$ cm²/s

Watch out for this

Putting numbers in before differentiating.

If $x = 6$ is fixed too soon, its rate disappears. Differentiate first.

The short version
Write an equation linking the quantities. Differentiate with respect to

t

. Substitute numbers only at the end.

Your turn

Prefer paper? Print the More derivatives and applications worksheet, with an answer key.

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Work through it

A 10 ft ladder's base slides out at 1 ft/s. How fast does the top fall when x=6?

A circle's radius grows at 2 cm/s. How fast is its area growing when r=5?

Watch out for this

Your turn

A 10 ft ladder's base slides out at 1 ft/s. How fast does the top fall when $x = 6$ ?

A circle's radius grows at 2 cm/s. How fast is its area growing when $r = 5$ ?