Implicit Differentiation

You’ll be able to

Differentiate equations that are not solved for y using implicit differentiation
Apply the chain rule to y-terms, attaching dy/dx
Solve for dy/dx and evaluate the slope at a given point

When you cannot solve for y

Some curves — like the circle x² + y² = 25 — cannot be written as a single function y = f(x). Implicit differentiation handles them by treating y as an unknown function of x and differentiating both sides of the equation with respect to x. The key move: every time you differentiate a term containing y, the chain rule attaches a factor of dy/dx, because y itself depends on x. Then you solve the resulting equation algebraically for dy/dx.

Differentiating a y-term

d/dx [ y³ ] = 3y² · (dy/dx) · d/dx [ x·y ] = y + x·(dy/dx)

A y raised to a power picks up dy/dx by the chain rule; a product of x and y needs the product rule too.

The four-step method

Implicit differentiation follows a reliable routine: (1) differentiate both sides term by term with respect to x, attaching dy/dx to every y-term; (2) collect all terms containing dy/dx on one side; (3) factor out dy/dx; (4) divide to isolate dy/dx. The answer typically involves both x and y, which is expected — to get a numerical slope you substitute a specific point (x, y) that lies on the curve.

Worked example

Find dy/dx for the circle x² + y² = 25, then the slope at the point (3, 4).

1.Differentiate both sides with respect to x: d/dx[x²] + d/dx[y²] = d/dx[25].
2.This gives 2x + 2y·(dy/dx) = 0, where the y² term picks up dy/dx by the chain rule.
3.Solve for the derivative: 2y·(dy/dx) = −2x, so dy/dx = −x/y.
4.Substitute the point (3, 4): dy/dx = −3/4.

Answer: dy/dx = −x/y, and at (3, 4) the slope of the tangent is −3/4.

Watch out

The most common implicit-differentiation error is differentiating a y-term as if y were a constant. d/dx[y²] is 2y·(dy/dx), not 2y. Every y carries a hidden dy/dx factor — forgetting it invalidates the whole solution.

Checkpoint

When differentiating the term y² with respect to x, what do you get?

Checkpoint

For the curve xy = 6, what is dy/dx?

On the exam

Implicit results usually contain both x and y — that is correct, not incomplete. To report a numerical slope you must plug in a full point (x, y) on the curve. Free-response graders expect the point substituted, not just the symbolic derivative.

Answer the 2 checkpoints as you read.