Implicit Differentiation

Implicit differentiation lets you find $dy/dx$ even when an equation does not isolate $y$. Instead of solving for $y$ first, you differentiate both sides with respect to $x$ and treat $y$ as a function of $x$.

What implicit differentiation means

Start with a relation such as

$$F(x,y) = 0$$

If that relation defines $y$ as a differentiable function of $x$ near the point you care about, then you can differentiate the whole equation with respect to $x$ and solve for $dy/dx$.

The main idea is simple:

1. Differentiate every term with respect to $x$.
2. Treat $y$ as changing with $x$.
3. Solve the new equation for $dy/dx$.

That second step is the one students usually miss. For example,

$$\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$$

not just $2y$.

Why you need it

Some curves are easy to describe with one equation but awkward to write as a single formula $y = f(x)$. A circle is the standard example:

$$x^2 + y^2 = 25$$

This equation represents the whole circle at once. Solving for $y$ would split it into a top branch and a bottom branch, but implicit differentiation lets you find the slope directly from the original relation.

Worked example: slope of a circle

Find $dy/dx$ for

$$x^2 + y^2 = 25$$

Differentiate both sides with respect to $x$:

$$\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(25)$$ $$2x + 2y \frac{dy}{dx} = 0$$

Now solve for $dy/dx$:

$$2y \frac{dy}{dx} = -2x$$ $$\frac{dy}{dx} = -\frac{x}{y}$$

This formula works at points on the circle where $y \ne 0$. If $y = 0$, then dividing by $y$ is not valid, and on this circle those points correspond to vertical tangents.

At the point $(3,4)$,

$$\frac{dy}{dx} = -\frac{3}{4}$$

so the tangent line slopes downward there.

Where the chain rule appears

The chain rule enters whenever you differentiate a term containing $y$, because $y$ depends on $x$.

For instance,

$$\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}$$

and

$$\frac{d}{dx}(\sin y) = \cos(y)\frac{dy}{dx}$$

If you do not see any $dy/dx$ term after differentiating an expression that contained $y$, stop and check that step again.

Common mistakes in implicit differentiation

1. Differentiating $y^2$ as $2y$ instead of $2y \frac{dy}{dx}$.
2. Forgetting that a mixed term such as $xy$ needs the product rule.
3. Solving for $dy/dx$ by dividing by an expression without checking whether it could be $0$.
4. Assuming one derivative formula works globally, even when the relation has multiple branches.

When implicit differentiation is used

Implicit differentiation is most useful when:

1. A curve is given by a relation such as a circle, ellipse, or level curve.
2. Solving explicitly for $y$ would be messy or would split the curve into separate cases.
3. You need the slope of a tangent line at a point.
4. A related-rates problem links changing variables before you differentiate with respect to time.

Try a slightly harder example

Try

$$x^2 + xy + y^2 = 7$$

Differentiate both sides and solve for $dy/dx$. This is a good check because the term $xy$ requires the product rule, while $y^2$ still produces a chain rule factor.

If you want a next step, try your own version with a mixed term and then compare it to the product rule and chain rule cases separately.