Short answer
Implicit differentiation treats y as a function of x. Differentiate both sides with respect to x, apply chain rule to y terms, then isolate dy/dx.
- Every derivative of a y-term needs a dy/dx factor.
- Differentiate both sides term by term before solving.
- Substitute specific points only after isolating dy/dx.
Differentiate mixed-variable equations
In equations like x^2 + y^2 = 25, y is not isolated, but it still depends on x. Derivative of y^2 is 2y*(dy/dx), not just 2y.
Product and chain rules can appear together when terms include x and y multiplied.
Solve algebraically for dy/dx
After differentiation, group terms containing dy/dx on one side and move the rest to the other side.
Factor dy/dx and divide. Keep signs and parentheses explicit to avoid algebra mistakes.
- Mark dy/dx in every y-derived term.
- Factor dy/dx once, then isolate.
- Check with geometry when possible, such as tangent slope on a circle.
Find slope on a circle
Given x^2 + y^2 = 25, find dy/dx and then slope at (3,4).
- Differentiate both sides: 2x + 2y*(dy/dx) = 0.
- Isolate dy/dx: 2y*(dy/dx) = -2x, so dy/dx = -x/y.
- Substitute (3,4): slope = -3/4.
Result: The tangent slope at (3,4) is -3/4.
Common mistakes
- Differentiating y^n without multiplying by dy/dx.
- Substituting point values before isolating dy/dx.
- Dropping terms during algebra rearrangement.
- Assuming y is constant during differentiation.
Try one
Why does d/dx(y^3) equal 3y^2*(dy/dx) in implicit work?
Because y depends on x, so chain rule adds the factor dy/dx.
Sources
- OpenStax Calculus Volume 1Open textbook covering limits, derivatives, applications, and introductory integration.
- College Board AP CalculusOfficial AP Calculus AB course overview and unit structure.