# Find the Horizontal Tangent Line x^2+y^2=169

Solve the equation as in terms of .
Subtract from both sides of the equation.
Take the square root of both sides of the equation to eliminate the exponent on the left side.
The complete solution is the result of both the positive and negative portions of the solution.
Simplify the right side of the equation.
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
The complete solution is the result of both the positive and negative portions of the solution.
First, use the positive value of the to find the first solution.
Next, use the negative value of the to find the second solution.
The complete solution is the result of both the positive and negative portions of the solution.
Set each solution of as a function of .
Because the variable in the equation has a degree greater than , use implicit differentiation to solve for the derivative .
Differentiate both sides of the equation.
Differentiate the left side of the equation.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
Rewrite as .
Reorder terms.
Since is constant with respect to , the derivative of with respect to is .
Reform the equation by setting the left side equal to the right side.
Solve for .
Subtract from both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Simplify .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Simplify .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Move the negative in front of the fraction.
Replace with .
Set the derivative equal to then solve the equation .
Multiply each term in by
Multiply each term in by .
Multiply .
Multiply by .
Multiply by .
Multiply by .
Multiply both sides of the equation by .
Multiply by .
Solve the function at .
Replace the variable with in the expression.
Simplify the result.
Remove parentheses.
Multiply by .
Multiply by .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Solve the function at .
Replace the variable with in the expression.
Simplify the result.
Remove parentheses.
Multiply by .