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What does implicit differentiation actually do

Implicit differentiation finds the derivative of a variable defined implicitly by an equation, by differentiating both sides with respect to the independent variable and solving for the desired derivative. It lets you compute slopes even when y cannot be isolated explicitly.

Calculus · Derivatives

When an equation ties x and y together, such as x2x^2+ y2y^2 = 25, y is not expressed as a function of x. Implicit differentiation treats y as an unknown function of x, differentiates every term using the chain rule, and then isolates dy/dx. This technique works for algebraic, trigonometric, and exponential relationships where solving for y first would be messy or impossible.

Why you can't just solve for y

Key points about implicit differentiation

  • You differentiate both sides with respect to x, not y.
  • Every occurrence of y is treated as y(x) and thus gains a factor dy/dx when differentiated.
  • The resulting equation is linear in dy/dx, making it easy to solve.
  • It works for curves like circles, ellipses, and implicit trig identities.

Procedure for an implicit derivative

  1. 1Write the original equation involving x and y.
  2. 2Differentiate each term with respect to x, applying the chain rule to y‑terms (multiply by dy/dx).
  3. 3Collect all dy/dx terms on one side of the equation.
  4. 4Factor out dy/dx if necessary.
  5. 5Solve for dy/dx to obtain the derivative.

Explicit vs implicit differentiation

MethodDerivative of y
Explicit: y = x2x^2dy/dx = 2x
Implicit: x2x^2 + y2y^2 = 25dy/dx = -x/y

Check yourself

When differentiating the circle x2x^2 + y2y^2 = 25 implicitly, what is dy/dx at the point (3,4)?

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