# Find the Horizontal Tangent Line f(x)=x^3-7x^2

Find the derivative.
Differentiate.
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Set the derivative equal to then solve the equation .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Set the first factor equal to .
Set the next factor equal to and solve.
Set the next factor equal to .
Add to both sides of the equation.
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
The final solution is all the values that make true.
Solve the original function at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Raising to any positive power yields .
Raising to any positive power yields .
Multiply by .
Solve the original function at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Apply the product rule to .
Raise to the power of .
Raise to the power of .
Multiply .
Combine and .
Multiply by .
Move the negative in front of the fraction.
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply and .
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.