Implicit Differentiation

Summary

When a relation $F(x,y)=C$ defines $y$ as a function of $x$ only implicitly, differentiate both sides with respect to $x$ and solve for $y^{\prime }$ . Use the chain rule on every term involving $y$ .

Prerequisites

Derivatives, Chain Rules

Procedure

1. Differentiate each term of $F(x,y)=C$ with respect to $x$ , treating $y=y(x)$ .
2. Collect all terms that contain $y^{\prime }$ .
3. Solve linearly for $y^{\prime }$ .

For a level set $F(x,y)=C$ with $F_y\ne 0$ ,

Conditions / Assumptions

• Local solvability for $y$ as a $C^1$ function of $x$ requires $F_y\ne 0$ at the point (implicit function theorem).
• At points where $F_y=0$ and $F_x\ne 0$ , the tangent may be vertical.

Worked Example

Circle

From $x^2+y^2=100$ ,

Exponential relation (correct isolation)

From $e^{xy}+x^2-y^2=5$ ,

Collect $y^{\prime }$ terms:

so

provided the denominator is nonzero. Equivalently, $y^{\prime }=-F_x/F_y$ with $F=e^{xy}+x^2-y^2$ .

Implicit partial derivatives

For $x^3+y^3+z^3+6xyz=1$ with $z=z(x,y)$ ,

when $z^2+2xy\ne 0$ .

Common Mistakes

• Forgetting the chain-rule factor $y^{\prime }$ when differentiating $y^2$ , $e^{xy}$ , etc.
• Incorrect algebra when isolating $y^{\prime }$ (must move all $y^{\prime }$ terms to one side before dividing).
• Using $-F_x/F_y$ without checking $F_y\ne 0$ .

Connections

• Related formula for level curves: Chain Rules
• Surfaces: Tangent Plane

References

Implicit differentiation is developed in OpenStax Calculus Volume 1; the multivariable form uses the implicit function theorem setup in Volume 3.^[1]