Implicit differentiation

IB Mathematics Analysis and Approaches HL· AA HL 5.4 Differentiation techniques· 15 min read

1. Implicit Differentiation★★★☆☆⏱ 5 min

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📘 Definition

Implicit Differentiation

The process of differentiating both sides of an implicit relation with respect to $x$ , applying the chain rule to all terms containing $y$ .

Example:

Used for relations like $x^{2} + y^{3} = x y$ that are hard to rearrange to $y = f (x)$ .

By the chain rule, any function of $y$ differentiated with respect to $x$ gives the identity:

\frac{d}{dx}\left[f(y)\right] = f'(y) \cdot \frac{dy}{dx}

📐 Worked Example

Find $\frac{d y}{d x}$ for $x^{2} + y^{2} = 25$ , then calculate the gradient at $(3, 4)$ .

1
Differentiate each term on both sides with respect to $x$ :
2
$2x + 2y \cdot \frac{dy}{dx} = 0$
3
Rearrange to isolate $\frac{d y}{d x}$ :
4
$2y \frac{dy}{dx} = -2x \implies \frac{dy}{dx} = -\frac{x}{y}$
5
Substitute the point $(3, 4)$ to get the gradient:
6
$\frac{dy}{dx}\bigg|_{(3,4)} = -\frac{3}{4}$

Exam tip:

Always remember the chain rule factor for terms containing $y$ .

2. Common Pitfalls

Wrong move:

Forgetting to multiply by $\frac{d y}{d x}$ after differentiating a $y$ -term in implicit differentiation.

Why:

You are differentiating with respect to $x$ , not $y$ , so the chain rule requires the extra factor.

Correct move:

Add $\frac{d y}{d x}$ as a factor every time you differentiate a function of $y$ .

3. Quick Reference Cheatsheet

Technique	Core Formula	Use Case
Implicit Differentiation	Differentiate term-by-term: $\frac{d}{d x} [f (y)] = f^{'} (y) \frac{d y}{d x}$ , then rearrange	For relations that can't be written as $y = f (x)$

Going deeper

VideoImplicit differentiation worked examplesExtra practice for core concepts

What's Next

Mastering implicit differentiation is essential for later AA HL calculus. You will next apply it to find tangents, normals and stationary points on implicitly-defined curves, and it underpins related rates of change.