Topic 3.2 Notes – Implicit Differentiation

What Implicit Differentiation Is

An explicit equation looks like $y = x^3 - 4x$. You can just differentiate directly.

An implicit equation mixes $x$ and $y$ together, like
$$x^2 + xy + y^2 = 7.$$

Here, $y$ depends on $x$, but it’s not isolated. Instead of solving for $y$ (which could be messy or impossible), we differentiate both sides with respect to $x$.

The key idea:

• Treat $y$ as a function of $x$.
• Every time you differentiate a term with $y$, multiply by $\frac{dy}{dx}$.
• This is just the chain rule in disguise.

For example:
$$\frac{d}{dx}(y^4) = 4y^3 \frac{dy}{dx}.$$

You are differentiating the outside first, then multiplying by the derivative of the inside, which is $\frac{dy}{dx}$.

You’ll usually write the derivative as $\frac{dy}{dx}$ or $y'$. Stick with that notation.

The Process for Finding $\frac{dy}{dx}$

When you see an implicit equation, the steps are mechanical.

1. Differentiate both sides with respect to $x$
   Apply derivative rules to every term.
2. Use the chain rule for all $y$-terms
   Attach $\frac{dy}{dx}$ every time you differentiate something involving $y$.
3. Collect all $\frac{dy}{dx}$ terms on one side
4. Factor out $\frac{dy}{dx}$
5. Solve algebraically

Let’s try one:

$$x^3 + y^3 = 6xy.$$

Differentiate both sides:

• $\frac{d}{dx}(x^3) = 3x^2$
• $\frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}$
• For $6xy$, use product rule:
  $$6\left(x\frac{dy}{dx} + y\right)$$

So we get:

$$3x^2 + 3y^2\frac{dy}{dx} = 6x\frac{dy}{dx} + 6y.$$

Group the $\frac{dy}{dx}$ terms:

$$3y^2\frac{dy}{dx} - 6x\frac{dy}{dx} = 6y - 3x^2.$$

Factor:

$$\frac{dy}{dx}(3y^2 - 6x) = 6y - 3x^2.$$

Solve:

$$\frac{dy}{dx} = \frac{6y - 3x^2}{3y^2 - 6x}.$$

Notice the final answer contains both $x$ and $y$. That’s normal.

Rules You Must Apply Correctly

Implicit differentiation still uses all the usual derivative rules.

• Power rule with $y$
  $\frac{d}{dx}(y^n) = n y^{n-1} \frac{dy}{dx}$
• Product rule
  $\frac{d}{dx}(xy) = x\frac{dy}{dx} + y$
• Trig functions
  $\frac{d}{dx}(\sin y) = \cos y \frac{dy}{dx}$
• Exponential
  $\frac{d}{dx}(e^y) = e^y \frac{dy}{dx}$

If it would require the chain rule normally, it still does here.

Geometry and Tangent Lines

Implicit differentiation often appears with curves like circles or ellipses.

For example:
$$x^2 + y^2 = 25$$

This circle has radius 5 and is centered at the origin. Suppose we want the slope of the tangent line at the point $(3,4)$.

Tangent line to $x^2 + y^2 = 25$ at $(3,4)$

Differentiate both sides with respect to $x$:

$$2x + 2y\frac{dy}{dx} = 0.$$

Solve for $\frac{dy}{dx}$:

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

Now evaluate at $(3,4)$:

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

Use point-slope form to write the tangent line:

$$y - 4 = -\frac{3}{4}(x - 3).$$

On FRQs, you often:

• Find $\frac{dy}{dx}$
• Evaluate it at a point
• Write the tangent line

Be careful to plug the point into the derivative expression, not the original equation.

When You Use This

Implicit differentiation is helpful when:

• Solving for $y$ would create messy radicals
• The equation represents multiple branches (like a circle)
• The algebra to isolate $y$ would waste time

The AP likes equations you could solve but shouldn’t. Implicit differentiation is faster and cleaner.

Common Mistakes

• Forgetting to multiply by $\frac{dy}{dx}$ on $y$-terms
• Missing product rule on mixed terms like $x\sin y$
• Trying to plug numbers in before solving for $\frac{dy}{dx}$
• Losing track of algebra when isolating $\frac{dy}{dx}$

Most errors are algebra mistakes, not calculus mistakes.