8 min readPathway sheet2-mark and 3-mark questions; also a step inside many longer answers

Derivatives of implicit functions

When x and y are mixed in one equation, differentiate every term with respect to x, adding dy/dx after each y-term, then solve for dy/dx.

Explicit vs implicitExplicit: y = f(x) is given directly. Implicit: an equation such as x³ + y³ = 9xy links x and y

Key stepd/dx (yⁿ) = nyⁿ⁻¹ × dy/dx, because y depends on x

Product termsd/dx (xy) = y + x dy/dx

Form of the answerdy/dx may contain both x and y

When to use it

Sometimes y cannot easily be written as a formula in x, as in x³ + y³ = 9xy. Such an equation defines y implicitly. You do not need to solve for y. Differentiate both sides with respect to x, treating y as a function of x. Each time you differentiate a term containing y, the chain rule adds a factor dy/dx.

Method

1
Differentiate every term on both sides with respect to x.
2
For each y term, write dy/dx after it: d/dx (y²) = 2y dy/dx and d/dx (sin y) = cos y dy/dx.
3
Use the product rule for terms such as xy or x sin y.
4
Collect all the dy/dx terms on one side and the rest on the other.
5★
Take dy/dx out as a common factor and divide.

★ key step

Example 1: a quadratic curve

Problem

Find dy/dx if x² − 3xy + 2y² = 5.

Working

Differentiate: 2x − 3(y + x dy/dx) + 4y dy/dx = 0
2x − 3y − 3x dy/dx + 4y dy/dx = 0
dy/dx (4y − 3x) = 3y − 2x
dy/dx = (3y − 2x)/(4y − 3x)

Answer

dy/dx = (3y − 2x)/(4y − 3x), where 4y ≠ 3x

Example 2: with trigonometric terms

Problem

Find dy/dx if x sin y + y cos x = 1.

Working

d/dx (x sin y) = sin y + x cos y dy/dx
d/dx (y cos x) = cos x dy/dx − y sin x
So sin y + x cos y dy/dx + cos x dy/dx − y sin x = 0
dy/dx (x cos y + cos x) = y sin x − sin y
dy/dx = (y sin x − sin y)/(x cos y + cos x)

Answer

dy/dx = (y sin x − sin y)/(x cos y + cos x)

Example 3: slope at a point

Problem

For the curve x³ + y³ = 9xy, find dy/dx at the point (2, 4).

Working

Check the point: 8 + 64 = 72 and 9 × 2 × 4 = 72, so it lies on the curve.
Differentiate: 3x² + 3y² dy/dx = 9(y + x dy/dx)
dy/dx (3y² − 9x) = 9y − 3x²
Divide top and bottom by 3: dy/dx = (3y − x²)/(y² − 3x)
At (2, 4): dy/dx = (12 − 4)/(16 − 6) = 8/10

Answer

dy/dx = 4/5 at (2, 4)

In real life

Many real curves, such as the outline of a cooling tower at a power plant, come as one equation in x and y together. Engineers find their slope by implicit differentiation, without solving for y.

Exam angle

2 marks: “Find dy/dx if 3x − 2y = cos y.” 3 − 2 dy/dx = −sin y dy/dx. So dy/dx (2 − sin y) = 3, and dy/dx = 3/(2 − sin y).

2 marks: “If x² + 2xy + y³ = 29, find dy/dx at the point (3, 2).” 2x + 2y + 2x dy/dx + 3y² dy/dx = 0, so dy/dx = −(2x + 2y)/(2x + 3y²). At (3, 2): −10/18 = −5/9.

3 marks: “Find dy/dx if sin(x + y) = xy.” cos(x + y)(1 + dy/dx) = y + x dy/dx. So dy/dx [cos(x + y) − x] = y − cos(x + y), and dy/dx = [y − cos(x + y)]/[cos(x + y) − x].

Remember it

Touch a y, write a dy/dx.