Rate of change

A derivative $\dfrac{dy}{dx}$ is the instantaneous rate at which $y$ changes as $x$ changes. When $x$ is time, $\dfrac{dy}{dt}$ is a velocity, growth rate, or flow rate. When two quantities both depend on time and are related by an equation, the chain rule connects their rates , the related rates problem.

Single variable rate

If $y = f(x)$ and $x$ varies with time, then

$\frac{dy}{dt} = f'(x) \cdot \frac{dx}{dt}.$

So $f'(x)$ is the conversion factor between rates.

Worked example: a spreading oil slick

An oil slick is a growing circle with radius $r$ increasing at $2$ cm/s. Find the rate at which the area $A = \pi r^2$ is growing when $r = 5$.

$\dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt} = 2\pi \cdot 5 \cdot 2 = 20 \pi$ cm$^2$/s.

Related rates: the procedure

1. Identify all variables, sketch if helpful.
2. Write the equation relating the variables.
3. Differentiate both sides with respect to time $t$, using the chain rule.
4. Substitute the given values to find the desired rate.

Worked example: a ladder sliding down a wall

A $10$-m ladder leans against a wall. The bottom slides away from the wall at $0.5$ m/s. How fast is the top sliding down when the bottom is $6$ m from the wall?

Let $x$ = horizontal distance, $y$ = vertical distance. $x^2 + y^2 = 100$.

Differentiate: $2x\dot x + 2y\dot y = 0$, so $\dot y = -\dfrac{x \dot x}{y}$.

At $x = 6$: $y = 8$. $\dot y = -\dfrac{6 \cdot 0.5}{8} = -3/8$ m/s.

The top is sliding down at $3/8$ m/s.

Worked examples

Example 1. A balloon's volume $V = \dfrac{4}{3}\pi r^3$ increases at $20$ cm$^3$/s. Find the rate of change of radius when $r = 2$ cm.

$\dfrac{dV}{dt} = 4\pi r^2 \dfrac{dr}{dt}$, so $20 = 4\pi(4)\dfrac{dr}{dt}$, giving $\dfrac{dr}{dt} = \dfrac{5}{4\pi}$ cm/s.

Example 2. The radius of a circle is increasing at $3$ cm/s. Find the rate of change of its area when $r = 10$ cm.

$\dfrac{dA}{dt} = 2\pi r \dfrac{dr}{dt} = 60\pi$ cm$^2$/s.

Example 3. Water is poured into a conical tank (apex down) of height $10$ m and base radius $5$ m at $2$ m$^3$/min. How fast is the water level rising when the depth is $4$ m?

By similar triangles, water radius $r = h/2$. Volume $V = \tfrac{1}{3}\pi r^2 h = \tfrac{\pi h^3}{12}$. Differentiate: $\dfrac{dV}{dt} = \tfrac{\pi h^2}{4}\dfrac{dh}{dt}$. Substitute: $2 = \tfrac{\pi \cdot 16}{4}\dot h$, so $\dot h = 1/(2\pi)$ m/min.

Example 4. A particle moves along $y = x^2 + 1$. Find the rate of change of $y$ when $x = 2$ and $\dfrac{dx}{dt} = 5$.

$\dfrac{dy}{dt} = 2x \dfrac{dx}{dt} = 2 \cdot 2 \cdot 5 = 20$.

Example 5. The side of a square is increasing at $0.5$ cm/s. Find the rate of increase of the diagonal when the side is $4$ cm.

Diagonal $d = s\sqrt 2$. $\dfrac{dd}{dt} = \sqrt 2 \dfrac{ds}{dt} = \dfrac{\sqrt 2}{2} = \dfrac{1}{\sqrt 2}$ cm/s. (Independent of side length.)

Example 6. Total revenue from sale of $x$ units is $R(x) = 13 x^2 + 26 x + 15$. Find the marginal revenue at $x = 7$.

Marginal revenue $= R'(x) = 26 x + 26$. At $x = 7$: $26 \cdot 7 + 26 = 208$.

Try it yourself

1. The radius of a sphere is increasing at $0.5$ cm/s. Find the rate of change of volume when $r = 4$ cm.
2. A man $2$ m tall walks away from a $6$-m lamp post at $1.5$ m/s. Find the rate at which his shadow lengthens.
3. The side of an equilateral triangle increases at $2$ cm/s. Find the rate of change of area when the side is $10$ cm.
4. A cylindrical tank of radius $3$ m has water rising at $0.5$ m/min. Find the volume flow rate.
5. The radius of a circular ripple in water increases at $4$ cm/s. Find the rate of growth of the area when $r = 6$ cm.
6. Total cost $C(x) = 0.007 x^3 - 0.003 x^2 + 15 x + 4000$. Find the marginal cost at $x = 17$.
7. Find the rate of change of the surface area of a sphere when the radius is $5$ and growing at $2$ cm/s.
8. A particle moves so that $s(t) = t^3 - 6t^2 + 9t + 5$. Find the velocity and acceleration at $t = 2$.
9. Volume of a cube increases at $9$ cm$^3$/s. How fast is the surface area increasing when the edge is $10$ cm?
10. The volume of a sphere is increasing at $8\pi$ cm$^3$/s. Find the rate of change of its surface area when $r = 5$.
11. A kite is at height $80$ m flying horizontally at $5$ m/s. How fast is the string being released when its length is $100$ m?
12. The radius of a balloon is decreasing at $1$ cm/s. Find the rate of change of volume when $r = 3$ cm.
13. A ladder of length $13$ m leans against a wall. Its foot is being pulled away at $2$ cm/s. Find the rate at which the top slides down when the foot is $5$ m from the wall.
14. A trough is in the shape of an inverted prism. Water flows in at $2$ m$^3$/min. Find the rate of rise of water when depth is $1$ m if the cross-section is an equilateral triangle of side $2$ m.

Pitfalls / Tricks

• Identify variables and rates first, write the constraint equation, then differentiate.
• Chain rule with respect to time: every variable that depends on $t$ contributes a $d/dt$ factor.
• Be careful with signs: rates can be negative (e.g., a ladder top sliding down).
• Always substitute given values after differentiating, never before.
• Units must be consistent , convert if necessary.

Next, monotonicity from the sign of $f'$.