What Implicit Differentiation Means | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

Treat y as a function

An explicit equation has already isolated $y$ . Here, $y$ remains a hidden function of $x$ , so the chain rule records how it changes.

Definition

What is meant by implicit differentiation?

Model answer: Differentiate both sides with respect to $x$ , while treating $y$ as a function of $x$ . Then solve the resulting equation for $dy/dx$ .

\frac{d}{dx}(y^n)=ny^{n-1}\frac{dy}{dx}

The factor $dy/dx$ is the chain rule: the outer power changes with $y$ , and $y$ changes with $x$ .

Every y-term carries y′

x² + y² = xy + 7

↓ differentiate every term

2x + 2y y′ = x y′ + y

xy → x y′ + y by the product rule

Treat y as a function of x even when the formula does not isolate y.

Differentiate the relationship

Finish the derivative rule before using a point. Substituting coordinates too early hides the general relationship.

Worked example

Gradient of an implicit circle

x^2+y^2=25

2x+2y\frac{dy}{dx}=0

\frac{dy}{dx}=-\frac{x}{y}

Point x-coordinate0

Point on the upper half: (0, 5)
Implicit gradient −x/y: 0

The blue curve satisfies x² + y² = 25. The dashed line uses the gradient found from implicit differentiation.

Your turnIndependent check [6 marks]

The curve $x^2+3y^2=28$ passes through $(1,3)$ . Find the equation of its tangent there in the form $ax+by+c=0$ .

Show worked answer

2x+6y\frac{dy}{dx}=0

\left.\frac{dy}{dx}\right|_{(1,3)}=-\frac19

Answer

x+9y-28=0