Implicit Differentiation
In short: Implicit differentiation obtains dy/dx straight from an equation that mixes x and y, by taking the derivative of every term in x and treating y as an unknown function of x.
AP Calculus ABCalculusAP Calculus AB Unit 3
What is Implicit Differentiation?
Implicit differentiation delivers slopes for curves that refuse to be written as one formula \(y = f(x)\), a circle being the classic case. Differentiate each term of the equation in \(x\), and keep in mind that \(y\) quietly depends on \(x\).
That reminder carries a cost. Every time a term contains \(y\), the chain rule tacks on a factor \(\frac{dy}{dx}\); for example \(\frac{d}{dx}[y^2] = 2y\,\frac{dy}{dx}\). Finally, pull the \(\frac{dy}{dx}\) terms together and solve for them.
How to spot it
- The curve is given as an equation mixing x and y, like x^2 + xy + y^2 = 7, with no y alone.
- A question asks for the slope or tangent line at a point that lies on such a curve.
- Isolating y would force a plus-or-minus square root, or would not be possible at all.
- A tangent slope is wanted and the final answer may contain both x and y, which implicit differentiation allows.
The method
- Start implicit differentiation by taking the derivative of both sides in \(x\); each \(y\)-term gets a factor \(\frac{dy}{dx}\).
- Use the product rule on terms such as \(xy\), where both letters appear.
- Move every term with \(\frac{dy}{dx}\) to one side, factor it out, and divide.
- Substitute the \(x\)- and \(y\)-coordinates of the point to get a numerical slope.
Implicit Differentiation: solved example
Problem
Find the slope of the curve \(x^2 + xy + y^2 = 7\) at the point \((1, 2)\).
Solution
Check the point: \(1 + 2 + 4 = 7\). Differentiate each term with respect to \(x\): \(2x + \left(y + x\frac{dy}{dx}\right) + 2y\frac{dy}{dx} = 0\).
Gather: \(\frac{dy}{dx}(x + 2y) = -(2x + y)\), so \(\frac{dy}{dx} = -\frac{2x + y}{x + 2y}\).
At \((1, 2)\): \(-\frac{2 + 2}{1 + 4} = \) \(-\frac{4}{5}\).
Try it
Question 1
Find \(\frac{dy}{dx}\) for \(y^3 + 4x = 5y\).
Show the answer
\(3y^2 y^{\prime} + 4 = 5y^{\prime}\), so \(y^{\prime}(3y^2 – 5) = -4\) and \(y^{\prime} = \frac{-4}{3y^2 – 5} = \frac{4}{5 – 3y^2}\).
Question 2
On \(x^2 + y^2 = 169\), find the slope at \((5, -12)\).
Show the answer
\(\frac{dy}{dx} = -\frac{x}{y} = -\frac{5}{-12} = \frac{5}{12}\).
Implicit Differentiation vs. Chain Rule
The chain rule is the engine inside the method: it is what puts \(\frac{dy}{dx}\) next to every \(y\)-term. The method itself is the whole procedure for an equation with x and y mixed together, including the algebra at the end to isolate \(\frac{dy}{dx}\).
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Whole dictionary: A–Z · AP Calculus AB · Calculus