Implicit Differentiation (AQA A-Level Mathematics): Flashcards

What is implicit differentiation?

Finding derivative when $y$ not explicitly solved for in terms of $x$

When to use implicit differentiation

Implicit equations, relations, circles (where $y$ not isolated)

$\frac{d}{dx}$ equivalent to

$\frac{d}{dy} \times \frac{dy}{dx}$

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

$2y \frac{dy}{dx}$

Steps for implicit differentiation

1. Diff both sides w.r.t. $x$, 2. Collect $\frac{dy}{dx}$ terms, 3. Solve for $\frac{dy}{dx}$

Rule applied when differentiating $y$ terms

Chain rule (introduces $\frac{dy}{dx}$)

$\frac{dy}{dx}$ for $x^2 + y^2 = 1$

$\frac{-x}{y}$

Applications of implicit differentiation

Tangent/normal lines, related rates, inverse function derivatives

$\frac{d}{dx}(xy)$ using product rule

$y + x\frac{dy}{dx}$

Finding stationary points implicitly

Set $\frac{dy}{dx} = 0$ and solve