The idea

Implicit differentiation

Differentiating a relation that mixes x and y — like x² + y² = 25 — without solving for y, by differentiating both sides with respect to x and treating y as a function of x (so d/dx of y² is 2y dy/dx).

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Pure · Differentiation

Implicit differentiation

Why it works

When y won't isolate

Not every curve can be tidied into

y = f(x)

. The circle

x^2 + y^2 = 25

would split into two half-curves

y = \pm\sqrt{25 - x^2}

; relations like

x^3 + y^3 = 6xy

resist rearranging at all. Implicit differentiation finds the gradient without ever isolating

y

The chain-rule tax

The one idea you need:

y

is still a function of

x

, even if you can't see the formula. So whenever you differentiate a term containing

y

, the chain rule fires and a

\frac{dy}{dx}

drops out. Differentiating

y^2

with respect to

x

\frac{d}{dx}\big(y^2\big) = 2y\,\frac{dy}{dx}

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