Year 12 Calculus: Implicit differentiation

Implicit differentiation is used when y is mixed into the equation and has not been isolated as $y = f (x)$ . You differentiate both sides with respect to x and remember to apply the chain rule whenever you differentiate a term containing y.

When differentiating

y

with respect to x, write

d y / d x

. That is the step most students forget.

Worked example

Problem: Find

d y / d x

x^{2} + y^{2} = 25

Differentiate both sides: $d (x^{2}) / d x + d (y^{2}) / d x = d (25) / d x$ .
This gives $2 x + 2 y (d y / d x) = 0$ .
Rearrange: $2 y (d y / d x) = - 2 x$ .
Divide by $2 y$ : $d y / d x = - x / y$ .

Answer:

d y / d x = - x / y

Why the method works

An implicit equation still defines a relationship between $x$ and $y$ , even if $y$ is not isolated. Because $y$ depends on $x$ , any derivative of a term involving $y$ must include $d y / d x$ . That is just the chain rule appearing in a less obvious form.

Worked example 2

Problem: Find

d y / d x

x y + y^{2} = 10

Differentiate both sides: $d (x y) / d x + d (y^{2}) / d x = 0$ .
Use the product rule on $x y$ : $x (d y / d x) + y$ .
Differentiate $y^{2}$ : $2 y (d y / d x)$ .
So the derivative equation is $x (d y / d x) + y + 2 y (d y / d x) = 0$ .
Group the $d y / d x$ terms: $(x + 2 y) (d y / d x) = - y$ .
Therefore $d y / d x = - y / (x + 2 y)$ .

Answer:

d y / d x = - y / (x + 2 y)

Common mistakes

Forgetting $d y / d x$ when differentiating any term containing $y$ .
Missing the product rule when $x$ and $y$ are multiplied together.
Stopping before rearranging to make $d y / d x$ the subject.

Implicit differentiation is really an extension of ordinary differentiation, not a separate world. The only difference is that the relationship between x and y is hidden inside one equation rather than written explicitly as $y = f (x)$ . Once that idea is clear, the method becomes much less intimidating. You are still differentiating term by term, still using familiar rules, and still rearranging algebraically at the end.

It is also worth remembering why this method is useful. Many important curves, especially circles and conics, are easier to define implicitly than explicitly. So implicit differentiation is not just a technical trick. It is the natural language for finding slopes on curves that are not conveniently written as single-output functions.

Revision focus

In calculus, one of the best habits is to decide what role the expression is playing before you touch a rule. Is it a product, a composite function, an implicitly defined curve, or an antiderivative problem? That classification step is not wasted time. It is the step that usually determines whether the next line of working will be short and accurate or long and unstable. Strong calculus work looks calm because the structure is identified early, not because the algebra is magically easier.