Key takeaway: Implicit differentiation finds the slope dy/dx of a curve given by an equation in x and y without solving for y, by differentiating both sides and treating y as a function of x.
Grade 12 – AP Calculus ABAP Calculus AB Unit 3Calculus
Definition of Implicit Differentiation
Implicit differentiation is the method for curves written as a relation, such as \(x^2+y^2-6x+4y=12\), where \(y\) is not alone on one side. Some relations cannot be solved for \(y\) in a single formula, yet their graphs still have slopes.
Assume \(y\) secretly depends on \(x\) and differentiate every term with respect to \(x\). Each term containing \(y\) gets an extra factor \(\frac{dy}{dx}\) from the chain rule. Then collect those terms and solve for the slope.
Formula and notation
A power of \(y\) differentiates as
\[ \frac{d}{dx}(y^n)=n\,y^{n-1}\,\frac{dy}{dx}. \]
Terms with only \(x\) behave as usual. The slope formula that results often contains both letters, so a point on the curve is needed to produce a number.
Key facts
- Every y-term needs a dy/dx factor from the chain rule; x-terms never do.
- A product such as xy needs the product rule, with the dy/dx factor on the y part.
- Confirm that the chosen point really satisfies the original equation.
- Implicit differentiation also works on explicit formulas, which makes it a flexible habit.
Implicit Differentiation example
Find the slope of \(x^2+y^2-6x+4y=12\) at \((7,1)\).
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Check: \(49+1-42+4=12\), so the point lies on the curve.
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Differentiate each term: \(2x+2y\,\frac{dy}{dx}-6+4\,\frac{dy}{dx}=0\).
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Group: \((2y+4)\,\frac{dy}{dx}=6-2x\), so \(\frac{dy}{dx}=\frac{3-x}{y+2}\).
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Evaluate at \(x=7\), \(y=1\): \(\frac{-4}{3}\).
The tangent slope at \((7,1)\) is \(-\frac{4}{3}\).
Common mistake
Watch out
A student differentiates \(y^2\) as \(2y\) and \(4y\) as \(4\), obtaining \(2x+2y-6+4=0\).
Why it is wrong
Since \(y\) changes with \(x\), both \(2y\) and \(4\) must be multiplied by \(\frac{dy}{dx}\). The faulty line has no slope symbol at all, so it cannot be solved for a slope. Implicit differentiation demands the factor on every y-term, and the slope at \((7,1)\) comes out as \(-\frac{4}{3}\).
Related terms and lessons
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