Implicit Differentiation

Finding dy/dx when y is not explicitly expressed as a function of x

SPEC 7.4 • DIFFERENTIATION

What is Implicit Differentiation?

Sometimes we have equations where y is not expressed explicitly as a function of x.

Explicit vs Implicit

Explicit: y = x² + 3x − 5 (y is isolated)

Implicit: x² + y² = 25 (y is mixed with x)

For implicit equations, we differentiate both sides with respect to x, treating y as a function of x.

Key Rule

When differentiating a term involving y, use the chain rule:

\frac{d}{d x} [f (y)] = f^{'} (y) \cdot \frac{d y}{d x}

Products of x and y

For terms like xy, we need the product rule:

Product Rule

\frac{d}{d x} [x y] = y + x \frac{d y}{d x}

Differentiate x (get 1) × y + x × differentiate y (get dy/dx)

More generally:

ExpressionDerivative w.r.t. x

x y

y + x \frac{d y}{d x}

x^{2} y

2 x y + x^{2} \frac{d y}{d x}

x y^{2}

y^{2} + 2 x y \frac{d y}{d x}

Tip

Remember: differentiate with respect to y, then multiply by dy/dx.

Worked Example01

Find dy/dx for the circle: x² + y² = 25

1

Differentiate both sides with respect to x:

2

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}] = \frac{d}{d x} [25]

3

2 x + 2 y \frac{d y}{d x} = 0

4

Rearrange:

2 y \frac{d y}{d x} = - 2 x

5

\frac{d y}{d x} = - \frac{x}{y}

Answer

\frac{d y}{d x} = - \frac{x}{y}

Worked Example02

Find dy/dx for: x² + xy + y² = 7

1

Differentiate each term:

\frac{d}{d x} [x^{2}] + \frac{d}{d x} [x y] + \frac{d}{d x} [y^{2}] = 0

2

2 x + (y + x \frac{d y}{d x}) + 2 y \frac{d y}{d x} = 0

(product rule for xy)

3

Collect dy/dx terms:

2 x + y + (x + 2 y) \frac{d y}{d x} = 0

4

(x + 2 y) \frac{d y}{d x} = - (2 x + y)

Answer

\frac{d y}{d x} = - \frac{2 x + y}{x + 2 y}

Worked Example03

Find dy/dx for: y = x/(x² + 1) implicitly, by first rewriting as y(x² + 1) = x

1

Rewrite:

y (x^{2} + 1) = x

, i.e.,

x^{2} y + y = x

2

Differentiate using product rule on x²y:

3

2 x y + x^{2} \frac{d y}{d x} + \frac{d y}{d x} = 1

4

Collect dy/dx:

(x^{2} + 1) \frac{d y}{d x} = 1 - 2 x y

5

\frac{d y}{d x} = \frac{1 - 2 x y}{x ^{2} + 1}

Answer

\frac{d y}{d x} = \frac{1 - 2 x y}{x ^{2} + 1}

(or substitute y back:

\frac{1 - x ^{2}}{( x ^{2} + 1 ) ^{2}}

)

Worked Example04

Find dy/dx for: sin(xy) = x

1

Differentiate LHS using chain rule:

2

\frac{d}{d x} [sin (x y)] = cos (x y) \cdot \frac{d}{d x} [x y]

3

Using product rule for xy:

cos (x y) \cdot (y + x \frac{d y}{d x}) = 1

4

Expand:

y cos (x y) + x cos (x y) \frac{d y}{d x} = 1

5

Rearrange:

x cos (x y) \frac{d y}{d x} = 1 - y cos (x y)

Answer

\frac{d y}{d x} = \frac{1 - y c o s ( x y )}{x c o s ( x y )}

Worked Example05

Find dy/dx for: eʸ = x + y

1

Differentiate both sides with respect to x:

2

e^{y} \frac{d y}{d x} = 1 + \frac{d y}{d x}

3

Collect dy/dx:

e^{y} \frac{d y}{d x} - \frac{d y}{d x} = 1

4

Factor:

(e^{y} - 1) \frac{d y}{d x} = 1

Answer

\frac{d y}{d x} = \frac{1}{e ^{y} - 1}

Worked Example06

The curve x²y + y³ = 10 passes through (1, 2). Find the equation of the tangent at this point.

1

Differentiate:

2 x y + x^{2} \frac{d y}{d x} + 3 y^{2} \frac{d y}{d x} = 0

2

(x^{2} + 3 y^{2}) \frac{d y}{d x} = - 2 x y

3

\frac{d y}{d x} = \frac{- 2 x y}{x ^{2} + 3 y ^{2}}

4

At (1, 2):

\frac{d y}{d x} = \frac{- 2 ( 1 ) ( 2 )}{1 + 12} = \frac{- 4}{13}

5

Tangent:

y - 2 = - \frac{4}{13} (x - 1)

Answer

4 x + 13 y = 30

Practice

Question 1: Find dy/dx for the ellipse: x² + 4y² = 16

Differentiate:

2 x + 8 y \frac{d y}{d x} = 0

8 y \frac{d y}{d x} = - 2 x

Answer

\frac{d y}{d x} = - \frac{x}{4 y}

Practice

Question 2: Find dy/dx for: cos(x + y) = y

Differentiate:

- sin (x + y) \cdot (1 + \frac{d y}{d x}) = \frac{d y}{d x}

Expand:

- sin (x + y) - sin (x + y) \frac{d y}{d x} = \frac{d y}{d x}

Collect:

- sin (x + y) = \frac{d y}{d x} + sin (x + y) \frac{d y}{d x} = \frac{d y}{d x} (1 + sin (x + y))

Answer

\frac{d y}{d x} = \frac{- s i n ( x + y )}{1 + s i n ( x + y )}

Practice

Question 3: The curve x³ + y³ = 6xy passes through (3, 3). Find the gradient at this point.

Differentiate:

3 x^{2} + 3 y^{2} \frac{d y}{d x} = 6 y + 6 x \frac{d y}{d x}

Collect:

3 y^{2} \frac{d y}{d x} - 6 x \frac{d y}{d x} = 6 y - 3 x^{2}

(3 y^{2} - 6 x) \frac{d y}{d x} = 6 y - 3 x^{2}

\frac{d y}{d x} = \frac{6 y - 3 x ^{2}}{3 y ^{2} - 6 x} = \frac{2 y - x ^{2}}{y ^{2} - 2 x}

At (3, 3):

\frac{d y}{d x} = \frac{6 - 9}{9 - 6} = \frac{- 3}{3} = - 1

AnswerGradient = −1

Summary

Implicit Differentiation Method

1

Differentiate both sides with respect to x

2

Apply chain rule to y terms: multiply by dy/dx

3

Apply product rule to xy terms

4

Apply chain rule to composite functions like sin(xy)

5

Collect all dy/dx terms and solve

Key Results

Curve Typedy/dxCircle: x² + y² = r²−x/yEllipse: x²/a² + y²/b² = 1−b²x/(a²y)Hyperbola: x² − y² = a²x/y

Common Mistakes

Forgetting dy/dx when differentiating y terms

Forgetting to use product rule for xy terms

Not applying chain rule correctly for trig/composite functions