slope of a curve - 2

Calculus Level 2

Find the slope of the curve x 2 + y 2 − 12 x + 4 y − 5 = 0 x^2+y^2-12x+4y-5=0 at point ( 0 , 1 ) (0,1) .

4 0 12 6 2

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1 solution

Relevant wiki: Implicit Differentiation Problem Solving - Basic

The derivative of a function is identical with the slope of the graph of the function. By implicit differentiation, we have

x 2 + y 2 − 12 x + 4 y − 5 = 0 x^2+y^2-12x+4y-5=0

2 x + 2 y y ′ − 12 + 4 y ′ = 0 2x+2yy'-12+4y'=0

2 y y ′ + 4 y ′ = 12 − 2 x 2yy'+4y'=12-2x

y ′ ( 2 y + 4 ) = 12 − 2 x y'(2y+4)=12-2x

y ′ = 12 − 2 x 2 y + 4 y'=\dfrac{12-2x}{2y+4}

When x = 0 x=0 and y = 1 y=1 , we have

y ′ = 12 − 2 ( 0 ) 2 ( 1 ) + 4 = 12 6 = 2 y'=\dfrac{12-2(0)}{2(1)+4}=\dfrac{12}{6}=\boxed{2}

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