AP Calculus AB and BC

Explicit vs Implicit Differentiation

Q: Why does dy/dx appear when I differentiate y?

Because y is a function of x, so the chain rule applies. Differentiating y^3 with respect to x gives 3y^2 \frac{dy}{dx}.

Differentiate explicitly when the equation is already solved for the output variable. Use implicit differentiation when the variables are mixed together and isolating one would be messy or impossible, as with a circle.

Explicit

Use when: The equation reads $y =$ something in $x$ alone, so you can differentiate directly.

Implicit

Use when: The equation mixes $x$ and $y$ , as in $x^2 + y^2 = 25$ or $xy + \sin y = 4$ .

Side by side

	Explicit	Implicit
Setup	$y = f(x)$	An equation in $x$ and $y$
Method	Differentiate the right side	Differentiate both sides, then solve for $\frac{dy}{dx}$
Result depends on	$x$ only	Usually both $x$ and $y$
Evaluating a slope needs	An $x$ value	A full point

The mechanical difference is one factor. Every time you differentiate a term containing $y$ , the chain rule contributes a $\frac{dy}{dx}$ , because $y$ is itself a function of $x$ . Differentiating $y^2$ gives $2y\frac{dy}{dx}$ , not $2y$ .

Because the answer usually contains both variables, evaluating an implicit derivative requires a complete point rather than just an $x$ value. That is also why implicit answers can describe curves that fail the vertical line test.

When explicit is still better

If the equation solves cleanly, solving first is often less work. Implicit differentiation earns its keep when solving would introduce square roots and cases, as it does for a circle.

Frequently asked questions

Why does dy/dx appear when I differentiate y?

Because $y$ is a function of $x$ , so the chain rule applies. Differentiating $y^3$ with respect to $x$ gives $3y^2 \frac{dy}{dx}$ .

In the CED: Unit 3: Chain Rule, Implicit, and Inverses