Find the Horizontal Tangent Line x^2+y^2=9

Problem

x2+y2=9

Solution

1. Differentiate implicitly with respect to x to find the slope of the tangent line.

d(x2)/d(x)+d(y2)/d(x)=d(9)/d(x)

2. Apply the chain rule to the term involving y

2*x+2*yd(y)/d(x)=0

3. Solve for the derivative d(y)/d(x) to represent the slope.

2*yd(y)/d(x)=−2*x

d(y)/d(x)=−x/y

4. Set the derivative to zero because a horizontal tangent line has a slope of 0

−x/y=0

x=0

5. Substitute the x-value back into the original equation to find the corresponding ycoordinates.

0+y2=9

y2=9

y=±3

6. Write the equations of the horizontal lines using the yvalues found.

y=3

y=−3

Final Answer

y=3,y=−3

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