A horizontal tangent line is a mathematical feature on a graph, located where a function's derivative is zero. This is because, by definition, the derivative gives the slope of the tangent line. Horizontal lines have a slope of zero. Therefore, when the derivative is zero, the tangent line is horizontal. To find horizontal tangent lines, use the derivative of the function to locate the zeros and plug them back into the original equation. Horizontal tangent lines are important in calculus because they indicate local maximum or minimum points in the original function.
Take the derivative of the function. Depending on the function, you may use the chain rule, product rule, quotient rule or other method. For example, given y=x^3 - 9x, take the derivative to get y'=3x^2 - 9 using the power rule that states taking the derivative of x^n, will give you n*x^(n-1).
Factor the derivative to make finding the zeros easier. Continuing with the example, y'=3x^2 - 9 factors to 3(x+sqrt(3))(x-sqrt(3))
Set the derivative equal to zero and solve for “x” or the independent variable in the equation. In the example, setting 3(x+sqrt(3))(x-sqrt(3))=0 gives x=-sqrt(3) and x=sqrt(3) from the second and third factors. The first factor, 3, doesn't give us a value. These values are the "x" values in the original function that are either local maximum or minimum points.
Plug the value(s) obtained in the previous step back into the original function. This will give you y=c for some constant “c.” This is the equation of the horizontal tangent line. Plug x=-sqrt(3) and x=sqrt(3) back into the function y=x^3 - 9x to get y= 10.3923 and y= -10.3923. These are the equations of the horizontal tangent lines for y=x^3 - 9x.