Implicit Differentiation Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

medium
Find the equation of the tangent line to x2y+y3=2x^2y + y^3 = 2 at the point (1,1)(1, 1).

Solution

  1. 1
    Differentiate: 2xy+x2dydx+3y2dydx=02xy + x^2\frac{dy}{dx} + 3y^2\frac{dy}{dx} = 0.
  2. 2
    Solve: dydx(x2+3y2)=−2xy⇒dydx=−2xyx2+3y2\frac{dy}{dx}(x^2 + 3y^2) = -2xy \Rightarrow \frac{dy}{dx} = \frac{-2xy}{x^2+3y^2}.
  3. 3
    At (1,1)(1,1): dydx=−24=−12\frac{dy}{dx} = \frac{-2}{4} = -\frac{1}{2}.
  4. 4
    Tangent line: y−1=−12(x−1)y - 1 = -\frac{1}{2}(x-1), i.e., y=−x2+32y = -\frac{x}{2} + \frac{3}{2}.

Answer

y=−x2+32y = -\frac{x}{2} + \frac{3}{2}
After finding dydx\frac{dy}{dx} implicitly, evaluate at the specific point to get the slope, then use point-slope form.

About Implicit Differentiation

Finding dydx\frac{dy}{dx} when yy is defined implicitly by an equation like F(x,y)=0F(x, y) = 0, by differentiating both sides and solving for dydx\frac{dy}{dx}.

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More Implicit Differentiation Examples

Example 1 easy

Find [formula] for the circle [formula] and evaluate it at the point [formula].

Example 2 hard

Find [formula] for [formula] (folium of Descartes).

Example 3 medium

Find [formula] for the circle [formula].

Example 4 easy

Find [formula] for [formula].

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