Third Derivatives of Implicit Curves

Why a Third Implicit Derivative Needs Care

An equation such as \(x^2+y^2=25\) can describe a curve without giving \(y\) explicitly as a formula in \(x\). At a point where the curve is locally a graph \(y=y(x)\), implicit differentiation lets us find its derivatives directly from the equation.

Finding a third derivative requires more than differentiating the original equation three times mechanically. After the first differentiation, \(y'\) is itself a quantity that changes with \(x\); after the second, so is \(y''\). In particular, the product rule can create several terms involving \(y'\) and \(y''\). Keeping them all is the central task.

For a polynomial equation \(F(x,y)=0\), the usual local condition that permits this treatment at a point is \(F_y\ne 0\), where \(F_y\) is the partial derivative with respect to \(y\). This is the implicit function theorem’s graph condition. If \(F_y=0\), the point may still lie on a curve, but this calculation does not establish that the curve is locally a differentiable graph \(y(x)\) there.

The Differentiation Pattern

When differentiating along a curve \(y=y(x)\), the chain rule gives \(\frac{d}{dx}F(x,y(x))=F_x+F_y y'\). The equation \(F(x,y(x))=0\) therefore gives a first-derivative relation. Differentiating that relation again produces a relation involving \(y''\); differentiating once more produces one involving \(y'''\).

The formula below records the third differentiation for a sufficiently differentiable function \(F\). Subscripts denote partial derivatives, and \(p=y'\), \(q=y''\), and \(r=y'''\) are used just to make the expression easier to read.

This formula is valid when the indicated derivatives exist and the curve is being differentiated as a function \(y(x)\). In examples, it is often less error-prone to differentiate the first- or second-derivative relation directly than to memorize the full formula.

Why the Third-Derivative Formula Has These Terms

Here is a derivation that also explains why terms involving \(y'y''\) must not be lost. Start with the second-derivative relation obtained by differentiating \(F_x+F_y y'=0\):

Differentiate each term with respect to \(x\), using the chain rule for each partial derivative. For instance, \(\frac{d}{dx}F_{xx}=F_{xxx}+F_{xxy}y'\). The four terms give:

Adding the expressions gives three copies of \(F_{xxy}y'\), three of \(F_{xyy}(y')^2\), three of \(F_{xy}y''\), and three of \(F_{yy}y'y''\). Grouping those terms yields the displayed third-derivative relation. The \(y'y''\) contribution comes from differentiating products; it cannot be discarded just because the original equation is polynomial.

Worked Examples

A Reliable Process

For a specific implicit equation, it is usually clearest to write down each differentiated relation before substituting the point’s coordinates. This keeps the algebraic derivatives separate from the numerical evaluation.

Common Mistake: Dropping a Product-Rule Term

A frequent error is to differentiate \(6y(y')^2\) as if \(y\) were constant, writing only \(12y y'y''\), or as if \((y')^2\) were constant, writing only \(6(y')^3\). Neither is complete: both factors depend on \(x\).

In the third worked example, omitting \(12yy'y''\) would also omit \(12yy'\) from the coefficient multiplying \(y''\) after the entire second-derivative relation is differentiated. At \((1,1)\), that missing contribution changes the equation and gives the wrong value of \(y'''\). The correction is to differentiate each factor explicitly before collecting like terms.

Why Third Implicit Derivatives Matter

The value \(y'''\) measures how the second derivative changes as \(x\) changes along the curve. It can distinguish curves whose positions, slopes, and concavity agree at a point but whose higher-order behavior differs. In calculation, its main lesson is practical: every derivative in an implicit relation is itself a function of \(x\), so another differentiation must account for its change.

Before accepting an answer, check that the point lies on the original curve, that the values of \(y'\) and \(y''\) satisfy their differentiated relations, and that the proposed \(y'''\) satisfies the third-derivative relation. These checks catch sign errors and missing product-rule terms.