Implicit Differentiation

What is Implicit Differentiation?

In many cases, we can easily find the derivative when a function is explicitly defined, such as

$$y=2x^2+6x+7$$

But sometimes, $y$ and $x$ are mixed together in one equation, for example:

$$x^2+y^2=9$$ or $$y^2+y = 3x^4-8x$$

Here, y is not written as a simple function of x.
To find $$\frac{dy}{dx}$$​, we use Implicit Differentiation.

The Idea

We differentiate both sides of the equation with respect to $x$,
and remember that y depends on x.

So whenever we differentiate a term containing y, we must multiply by $$\frac{dy}{dx}$$​ using the chain rule.

For example:

$$\frac{d}{dx}[y^2]=2y \cdot \frac{dy}{dx}$$​

because y is a function of x.

Step-by-Step Method

1. Differentiate both sides of the equation with respect to $x$.

2. Apply normal differentiation rules to $x$ terms.

3. Apply the chain rule to every term involving $y$.

4. Collect all $$\frac{dy}{dx}$$​ terms on one side.

5. Factor out $$\frac{dy}{dx}$$ and solve for it.

Example 1: Circle Equation

$$x^2+y^2=9$$

Differentiate both sides with respect to x:

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

Solve for $$\frac{dy}{dx}$$ :

$$\frac{-2x}{2y} = \frac{-x}{y}$$​

✅ Result: The slope of the tangent line to the circle $$x^2 + y^2 = 9$$ is $$\frac{-x}{y}$$​.

Example 2: $$x^3 + y^3 = 6xy$$

Differentiate both sides:

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

Now collect $$\frac{dy}{dx}$$ terms on one side:

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

Factor out $$\frac{dy}{dx}$$ :

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

Solve for $$\frac{dy}{dx}$$​:

$$\frac{dy}{dx} = \frac{6y – 3x^2}{3y^2 – 6x}$$​

✅ Simplify if needed:

​$$\frac{dy}{dx} = \frac{2y – x^2}{y^2 – 2x}$$​

Example 3: $$\sin(x+y) = y^2$$

Differentiate both sides:

$$\cos(x+y) \cdot (x+y)^\prime = 2y \frac{dy}{dx}$$​

$$\cos(x+y) \cdot (1+\frac{dy}{dx})^\prime = 2y \frac{dy}{dx}$$

Expand and collect $$\frac{dy}{dx}$$ terms:

$$\cos(x+y) + \cos(x+y) \frac{dy}{dx} = 2y \frac{dy}{dx}$$​

$$\cos(x+y) \frac{dy}{dx} – 2y \frac{dy}{dx} = -\cos(x+y)$$​

$$(\cos(x+y) – 2y) \frac{dy}{dx} = -\cos(x+y)$$​

$$\frac{dy}{dx} = \frac{-\cos(x+y)}{\cos(x+y) – 2y}= \frac{\cos(x+y)}{2y-\cos(x+y)}$$​

Why Use Implicit Differentiation?

Implicit differentiation is especially useful when:

• The equation mixes x and y (not easily solvable for y).

• You’re working with circles, ellipses, and curves.

• You need to find $$\frac{dy}{dx}$$ without isolating y.

Common Mistakes to Avoid

❌ Forgetting to multiply by $$\frac{dy}{dx}$$ when differentiating a $y$ term.
❌ Mixing up signs when moving terms across the equation.
❌ Not factoring $$\frac{dy}{dx}$$​ correctly.