Calculus I · Unit 2A · lesson

Second Derivatives from Implicit Equations

Concept

Learning objectives

Compute $d^2y/dx^2$ for implicit curves.

Differentiate the First-Derivative Relation Again

Explanation

Before the formulas

The key to Second Derivatives from Implicit Equations is to respect dependence that is present even when it is not written explicitly. If $y$ lies on a curve with $x$ , then $y$ changes when $x$ changes, and the chain rule produces a factor of $y'$ . Omitting that factor treats $y$ as a constant and changes the problem.

After differentiating, gather every term containing $y'$ on one side and factor it once. This is usually safer than moving terms randomly. Then solve and interpret the slope at a specified point, checking for denominator values that signal vertical tangent behavior.

Explanation

Differentiate the slope formula while remembering that $y$ still depends on $x$

A second implicit derivative often begins with a formula containing both $x$ and $y$ . Differentiating that formula requires product, quotient, and chain rules, and every new derivative of $y$ again introduces $dy/dx$ .

Substituting the first-derivative formula only after the second differentiation usually keeps the work organized. The result describes how tangent slope changes as you move along the implicit curve.

A second derivative of an implicit curve measures how its tangent slope changes along the curve. The calculation is more involved because the first derivative formula still contains both $x$ and $y$ , and differentiating it requires another round of implicit differentiation.

Keep $y'$ visible until the end. Substituting the first-derivative formula too early can inflate the algebra and hide the geometric structure you are trying to study.

To find $y"$ , differentiate an equation containing $y'$ with respect to $x$ . Products involving $y$ and $y'$ require the product rule, and the derivative of $y'$ is $y"$ .

Guided walkthrough

Second derivative of a circle

For $x^2+y^2=25$ , find $y"$ .

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

After the explanation

Use the section idea

Reading lens

Track which variable depends on which and use reciprocal or logarithmic structure only where its conditions hold.

Mental model

Implicit equations constrain variables together; inverse functions exchange inputs and outputs; logarithms turn products and powers into sums.

Decision

Choose implicit, inverse, or logarithmic differentiation from the equation's representation, not from surface complexity.

Common trap

Dropping a y-prime factor, using a reciprocal slope at the wrong point, or ignoring domain restrictions.

Check yourself

Can you identify the correspondence point and all hidden dependencies before differentiating?

Source & rights

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