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Calculus IChapter 3: DerivativesSection 3-10

Implicit Differentiation — Question 6

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Question 6

Consider the equation defined implicitly: x2+y2+ln⁡(y)=10x^2 + y^2 + \ln(y) = 10

  • (a) Use implicit differentiation to find dydx\dfrac{dy}{dx}.

  • (b) Find the slope of the tangent line at the point (3,1)(3, 1).

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Question 6 - Solution

The logarithm requires y>0y>0. Differentiating gives

2x+2yy′+y′/y=0,y′=−2x2y+1/y.2x+2yy'+y'/y=0,\qquad \boxed{y'=\frac{-2x}{2y+1/y}}.

At (3,1)(3,1), 32+12+ln⁡1=103^2+1^2+\ln1=10, so the point is on the curve.

The denominator is 33, giving y′(3)=−2\boxed{y'(3)=-2}.

Original worksheet page 2: question and worked solution for 3-10-006

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