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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 Differentiation
Edexcel A level Maths: 9.8 Implicit DifferentiationCurated by Kepler, not produced by us. Watch it 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
1

The key idea

Differentiate each term with respect to xx. Any term in yy gains a factor dydx\dfrac{dy}{dx} by the chain rule: ddx(y2)=2ydydx\dfrac{d}{dx}(y^2) = 2y\dfrac{dy}{dx}.

ddx f(y)=f′(y) dydx\frac{d}{dx}\,f(y) = f'(y)\,\frac{dy}{dx}
Chain rule on y-terms.
Example 1Differentiate x2+y2=25x^2 + y^2 = 25 implicitly.
12x+2ydydx=02x + 2y\tfrac{dy}{dx} = 0.
2dydx=−xy\tfrac{dy}{dx} = -\tfrac{x}{y}.
Answer
dydx=−xy\dfrac{dy}{dx} = -\dfrac{x}{y}

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

2

Products of x and y

A term like xyxy needs the product rule: ddx(xy)=y+xdydx\dfrac{d}{dx}(xy) = y + x\dfrac{dy}{dx}. After differentiating, collect the dydx\dfrac{dy}{dx} terms and solve.

Formula recap

ddx(yn)=nyn−1dydx\frac{d}{dx}(y^n) = n y^{n-1}\frac{dy}{dx}
Chain rule on y.
ddx(xy)=y+xdydx\frac{d}{dx}(xy) = y + x\frac{dy}{dx}
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.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. ddx(y2)\frac{d}{dx}(y^2) equals:

    • 2ydydx2y\frac{dy}{dx}
    • 2y2y
    • y2dydxy^2\frac{dy}{dx}
    • 2dydx2\frac{dy}{dx}
    Show answer and explanation

    Answer: 2ydydx2y\frac{dy}{dx}

    Chain rule.

  2. ddx(xy)\frac{d}{dx}(xy) equals:

    • y+xdydxy + x\frac{dy}{dx}
    • dydx\frac{dy}{dx}
    • x+yx + y
    • xydydxxy\frac{dy}{dx}
    Show answer and explanation

    Answer: y+xdydxy + x\frac{dy}{dx}

    Product rule.

  3. For x2+y2=25x^2 + y^2 = 25, dy/dx =

    • −xy-\frac{x}{y}
    • xy\frac{x}{y}
    • −yx-\frac{y}{x}
    • yx\frac{y}{x}
    Show answer and explanation

    Answer: −xy-\frac{x}{y}

    2x + 2y y′ = 0.

  4. Implicit differentiation treats y as:

    • a function of x
    • a constant
    • zero
    • equal to x
    Show answer and explanation

    Answer: a function of x

    Hence the chain rule.

  5. After differentiating you should:

    • collect dy/dx terms and solve
    • integrate
    • square both sides
    • set y = 0
    Show answer and explanation

    Answer: collect dy/dx terms and solve

    Make dy/dx the subject.

Next step

Keep practising

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