Implicit Differentiation Calculator

Implicit differentiation finds dy/dx when y is not solved for. Differentiate both sides with respect to x, multiplying every y-derivative by dy/dx. For the circle \(x^{2} + y^{2} = 25\): \(2x + 2y\frac{dy}{dx} = 0\), so \(\frac{dy}{dx} = -\frac{x}{y}\), and the slope at (3, 4) is −3/4.

How to use the implicit differentiation calculator

1. Enter an equation in x and y, such as x^2 + y^2 = 25.
2. Choose whether you also want the second derivative.
3. Optionally enter a point to get the slope there, the tangent line and the normal line slope.
4. Press Differentiate to see each step and the graph of the curve.

Method

1. Differentiate both sides with respect to x.
2. Every time you differentiate a term with y, multiply by \(\frac{dy}{dx}\) (the chain rule), because y depends on x.
3. Collect the \(\frac{dy}{dx}\) terms on one side and everything else on the other.
4. Factor out \(\frac{dy}{dx}\) and divide.

Shortcut: writing the equation as \(F(x, y) = 0\), \(\frac{dy}{dx} = -\frac{F_{x}}{F_{y}}\), where \(F_{x}\) and \(F_{y}\) are partial derivatives.

Worked example: folium of Descartes

\(x^{3} + y^{3} = 6xy\). Differentiate: \(3x^{2} + 3y^{2}\frac{dy}{dx} = 6y + 6x\frac{dy}{dx}\) (product rule on the right).

Collect: \((3y^{2} – 6x)\frac{dy}{dx} = 6y – 3x^{2}\), so \(\frac{dy}{dx} = \frac{2y – x^{2}}{y^{2} – 2x}\). At (3, 3) the slope is \(\frac{6 – 9}{9 – 6} = -1\), and the tangent line is y = −x + 6.

Second derivative

Differentiate \(\frac{dy}{dx}\) again with respect to x, still treating y as a function of x, then replace every \(\frac{dy}{dx}\) with its formula. For the circle: \(\frac{d^{2}y}{dx^{2}} = -\frac{x^{2}+y^{2}}{y^{3}} = -\frac{25}{y^{3}}\).

Common mistakes

• Forgetting \(\frac{dy}{dx}\) after differentiating \(y^{2}\) (it is \(2y\frac{dy}{dx}\), not \(2y\)).
• Forgetting the product rule on terms like xy.
• Evaluating at a point that is not on the curve.

Implicit differentiation practice quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

When do you use implicit differentiation?

When y is mixed into the equation with x and is hard or impossible to solve for, as in x³ + y³ = 6xy or sin(xy) = x.

Why do y terms get a dy/dx?

y is a function of x, so by the chain rule the derivative of g(y) with respect to x is g′(y)·dy/dx.

Can I get the second derivative?

Yes. Choose dy/dx and d²y/dx² and the calculator differentiates dy/dx again and substitutes dy/dx back in.

What if my point is not on the curve?

The calculator tells you, because the slope formula only describes the curve at points that satisfy the equation.

Is the answer always simplified?

The calculator picks the simplest equivalent form it finds and checks it numerically, but other equivalent forms are also correct.

Sources

• Paul’s Online Math Notes: Implicit Differentiation
• OpenStax Calculus Volume 1: Implicit Differentiation

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