AP Calculus AB — calculusStandard1 min read

Implicit Differentiation

Differentiate implicitly defined functions for AP Calculus AB.

Tutor AI Team
Tutor AI Team

Implicit differentiation handles equations where yy is not isolated, like circles and other curves.

Method

  1. Differentiate both sides with respect to xx.
  2. When differentiating yy terms, multiply by dydx\frac{dy}{dx}.
  3. Solve for dydx\frac{dy}{dx}.

Worked Example: Example 1

Problem

x2+y2=25x^2 + y^2 = 25. 2x+2ydydx=02x + 2y\frac{dy}{dx} = 0 → dydx=−xy\frac{dy}{dx} = -\frac{x}{y}.

Solution

Worked Example: Example 2

Problem

xy+y2=6xy + y^2 = 6. y+xdydx+2ydydx=0y + x\frac{dy}{dx} + 2y\frac{dy}{dx} = 0 → dydx=−yx+2y\frac{dy}{dx} = \frac{-y}{x + 2y}.

Solution

Worked Example: Tangent Line

Problem

Find tangent to x2+y2=25x^2 + y^2 = 25 at (3,4)(3, 4). dydx=−34\frac{dy}{dx} = -\frac{3}{4}. y−4=−34(x−3)y - 4 = -\frac{3}{4}(x - 3).

Solution

Practice Problems

    1. x3+y3=6xyx^3 + y^3 = 6xy. Find dydx\frac{dy}{dx}.
    1. Find the slope at (1,2)(1, 2) on x2y+xy2=12x^2y + xy^2 = 12.

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Key Takeaways

  • ✓

    Differentiate term by term, use chain rule on yy.

  • ✓

    Collect dydx\frac{dy}{dx} terms, solve.

  • ✓

    Evaluate at specific points for tangent lines.

Tags:
#ap#calculus#differentiation#implicit

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