Implicit Differentiation Math Example 1

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

Example 1

easy
Find dydx\frac{dy}{dx} for the circle x2+y2=25x^2 + y^2 = 25 and evaluate it at the point (3,4)(3, 4).

Solution

  1. 1
    Differentiate both sides with respect to xx: 2x+2ydydx=02x + 2y\frac{dy}{dx} = 0.
  2. 2
    Solve for dydx\frac{dy}{dx}: dydx=−xy\frac{dy}{dx} = -\frac{x}{y}.
  3. 3
    At (3,4)(3, 4): dydx=−34\frac{dy}{dx} = -\frac{3}{4}.

Answer

dydx=−xy\dfrac{dy}{dx} = -\dfrac{x}{y}; at (3,4)(3, 4): slope =−34= -\dfrac{3}{4}
Whenever a yy-term is differentiated, attach dydx\frac{dy}{dx} by the chain rule. Then collect all dydx\frac{dy}{dx} terms and solve. The tangent to a circle at (3,4)(3,4) has slope −3/4-3/4.

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 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].

Example 5 medium

Find the equation of the tangent line to [formula] at the point [formula].

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