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Differentiation methods

How to Do Implicit Differentiation: Steps and Examples

Implicit equations connect x and y without isolating y first. Treat y as a function of x: every differentiated y-term therefore carries y′, and products such as xy require the product rule.

Direct answer

Differentiate both sides, collect every y′ term, then solve for y′.

A local slope formula for y=y(x) requires the coefficient of y′ to be nonzero at the point being studied.

Worked method: x²+xy+y²=1

Differentiate both sides with respect to x. The product rule gives d(xy)/dx=y+xy′, and the chain rule gives d(y²)/dx=2yy′.

Collect the terms containing y′: xy′+2yy′=−2x−y. Factor y′ to obtain (x+2y)y′=−(2x+y).

Divide only where x+2y is nonzero: y′=−(2x+y)/(x+2y). A zero denominator means this local y-as-a-function-of-x formula is unavailable there; it does not automatically mean the original equation has no solution.

1
2x + y + xy′ + 2yy′ = 0
2
(x + 2y)y′ = −(2x + y)
3
y′ = −(2x + y)/(x + 2y), x+2y ≠ 0

Worked examples

x2+y2=25x^{2} + y^{2} = 25
y′ = −x/y

The local y(x) expression requires y≠0.

xy = 6
y′ = −y/x

Use the product rule: y+xy′=0, with x≠0.

y2=x3+1y^{2} = x^{3} + 1
y′ = 3x²/(2y)

Differentiating y² introduces 2yy′; the local formula requires y≠0.

Common mistakes

  • • Treating y as a constant even though the equation makes y depend on x.
  • • Writing d(xy)/dx as xy′ and omitting the y term from the product rule.
  • • Calling every point with a zero y′ coefficient “no solution”; the issue is the chosen local representation, not necessarily the curve itself.

Try the worked examples in the calculator

Each link opens the matching calculator with the expression, variable and required conditions already filled in.

Derivação implícita:x2+y2=25x^{2} + y^{2} = 25Derivação implícita:x2+x y+y2=1x^{2} + x\,y + y^{2} = 1
Open the calculator without a preset: Derivação implícita

References