Topic 4 of 7

Implicit differentiation and related rates

Step 15 · Calculus 1 topics

Differentiate equations that mix x and y without solving for y, and find how fast one quantity changes from how fast another does.

Implicit differentiation

Some curves, like x² + y² = 25, are not one function y = f(x). You can still find the slope: differentiate both sides with respect to x, treating y as a function of x. Each time you differentiate something with y in it, the chain rule adds a factor of $\frac{dy}{dx}$ .

Example: A circle

Find $\frac{dy}{dx}$ on x² + y² = 25, and the slope at (3, 4).

Differentiate: $2x + 2y\frac{dy}{dx} = 0$ .
Solve: $\frac{dy}{dx} = -\frac{x}{y}$ .
At (3, 4) this is $-\frac{3}{4}$ .

Answer $-\frac{3}{4}$

x² + y² = 25 with its tangent at (3, 4), slope −3/4. The radius to (3, 4) has slope 4/3, at right angles to the tangent.

Example: Products of x and y

Find $\frac{dy}{dx}$ for x³ + xy + y² = 7.

Differentiate, using the product rule on xy: $3x^{2} + y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0$ .
Gather the $\frac{dy}{dx}$ terms: $(x + 2y)\frac{dy}{dx} = -3x^{2} - y$ .

Answer $\frac{dy}{dx} = -\frac{3x^{2} + y}{x + 2y}$

Setting up related rates

In a related rates problem several quantities change with time, tied together by an equation. Differentiating that equation with respect to time t links their rates.

Draw a picture and name the quantities that change.
Write an equation that links them.
Differentiate both sides with respect to t.
Only then put in the numbers for the moment you care about.

Example: A growing ripple

A ripple's radius grows at 2 cm/s. How fast is its area growing when the radius is 10 cm?

A = πr².
Differentiate with respect to t: $\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$ .
Put in r = 10 and dr/dt = 2 to get 40π.

Answer40π, about 125.7 cm² per second

Solving related rates

Example: A sliding ladder

A 10 m ladder leans against a wall. Its foot slides away at 1 m/s. How fast does the top slide down when the foot is 6 m from the wall?

Let x be the foot's distance and y the top's height: x² + y² = 100.
Differentiate: $2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0$ .
When x = 6, y = 8, so 2(6)(1) + 2(8)(dy/dt) = 0.
$\frac{dy}{dt} = -\frac{12}{16} = -0.75$ .

AnswerDown at 0.75 m/s

Example: A shadow

A person 2 m tall walks away from a 6 m lamp post at 1.5 m/s. How fast does their shadow grow?

Let x be their distance from the post and s the shadow's length.
Similar triangles: 6 ÷ (x + s) = 2 ÷ s, so 6s = 2x + 2s and s is x ÷ 2.
So $\frac{ds}{dt} = \frac{1}{2}\frac{dx}{dt}$ , which is 0.75.

Answer0.75 m/s, however far away they are

Example: A cone filling

Water pours into a cone, point down, at 9 m³ per minute. The water's radius is always half its depth. How fast is the depth rising when it is 6 m?

$V = \frac{1}{3}\pi r^{2}h$ with r = h ÷ 2, so $V = \frac{\pi h^{3}}{12}$ .
$\frac{dV}{dt} = \frac{\pi h^{2}}{4}\frac{dh}{dt}$ , and at h = 6 that is 9π times dh/dt.
So 9 = 9π × dh/dt.

Answer $\frac{1}{\pi}$ , about 0.32 m per minute

Try it yourself

Work each one on paper first, then open the answer to check.

Find the slope of x² + y² = 100 at the point (6, 8).
Show answer
$-\frac{3}{4}$
Find $\frac{dy}{dx}$ for y³ = x.
Show answer
$\frac{1}{3y^{2}}$
Find $\frac{dy}{dx}$ for xy = 12.
Show answer
$-\frac{y}{x}$
A square's side grows at 3 cm/s. How fast is its area growing when the side is 5 cm?
Show answer
30 cm² per second
A balloon's radius grows at 1 cm/s. How fast is its volume growing when the radius is 3 cm?
Show answer
36π cm³ per second
A 13 m ladder's foot slides away from the wall at 2 m/s. How fast does the top slide down when the foot is 5 m out?
Show answer
$\frac{5}{6}$ m/s