M171 Calculus I · Unit 3: Implicit, limits & extrema · on Exam 3
§2.7 Implicit differentiation
Study notes for Implicit differentiation in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- When $y$ is defined by an equation like $x^2 + y^2 = 25$ rather than a formula $y = f(x)$, differentiate both sides with respect to $x$, treating $y$ as a function of $x$: every $y$-term picks up a factor $\frac{dy}{dx}$ by the chain rule.
- Then collect the $\frac{dy}{dx}$ terms on one side and solve. The answer usually involves both $x$ and $y$; to find a slope at a point, substitute both coordinates.
- Product terms like $xy$ need the product rule: $\frac{d}{dx}[xy] = y + x\frac{dy}{dx}$.
Key formulas
Chain on y
$$\frac{d}{dx}\big[y^n\big] = n y^{n-1}\frac{dy}{dx}$$
Circle
$$x^2+y^2=r^2 \;\Rightarrow\; \frac{dy}{dx} = -\frac{x}{y}$$
Worked example
Find the slope of $x^3 + y^3 = 9$ at $(1,2)$.
Show the solution
$3x^2 + 3y^2 y' = 0 \Rightarrow y' = -\dfrac{x^2}{y^2}$. At $(1,2)$: $y' = -\dfrac14$.
Common mistakes
- Forgetting $\frac{dy}{dx}$ on $y$-terms.
- Substituting the point before solving for y′ in the wrong spot, or substituting only x.
Exam tipBox every dy/dx as you produce it so none go missing when you collect terms.
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