9.8PureStretch

Implicit Differentiation

Some relations cannot be rearranged into y = f(x). Implicit differentiation lets you differentiate both sides with respect to x, treating y as a function of x via the chain rule.

26 min Video by Zeeshan Zamurred Differentiation

Edexcel A level Maths: 9.8 Implicit DifferentiationWatch the full walkthrough before the notes below.

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What you'll be able to do

Differentiate equations implicitly
Apply the chain rule to y-terms
Use the product rule on xy-terms
Make dy/dx the subject

The key idea

Differentiate each term with respect to $x$ . Any term in $y$ gains a factor $\frac{d y}{d x}$ by the chain rule: $\frac{d}{d x} (y^{2}) = 2 y \frac{d y}{d x}$ .

\frac{d}{d x} f (y) = f^{'} (y) \frac{d y}{d x}

Chain rule on y-terms.

2 x + 2 y \frac{d y}{d x} = 0

\frac{d y}{d x} = - \frac{x}{y}

Answer

\frac{d y}{d x} = - \frac{x}{y}

Tip — Every time you differentiate a y, tack on dy/dx.

Products of x and y

A term like $x y$ needs the product rule: $\frac{d}{d x} (x y) = y + x \frac{d y}{d x}$ . After differentiating, collect the $\frac{d y}{d x}$ terms and solve.

Formula recap

\frac{d}{d x} (y^{n}) = n y^{n - 1} \frac{d y}{d x}

Chain rule on y.

\frac{d}{d x} (x y) = y + x \frac{d y}{d x}

Product rule.

Common mistakes to avoid

Differentiating y² to 2y without dy/dx.

d/dx(y²) = 2y dy/dx.

Forgetting the product rule on xy.

d/dx(xy) = y + x dy/dx.

Key takeaways

Differentiate both sides w.r.t. x.
y-terms gain a dy/dx (chain rule); xy needs the product rule.
Collect dy/dx terms and make it the subject.

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