# Calculus Examples

Find the Horizontal Tangent Line y=2x^3+3x^2-12x+1
Step 1
Set as a function of .
Step 2
Find the derivative.
Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3
Multiply by .
Step 2.3
Evaluate .
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 2.4
Evaluate .
Step 2.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Multiply by .
Step 2.5
Differentiate using the Constant Rule.
Step 2.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.2
Step 3
Set the derivative equal to then solve the equation .
Step 3.1
Factor the left side of the equation.
Step 3.1.1
Factor out of .
Step 3.1.1.1
Factor out of .
Step 3.1.1.2
Factor out of .
Step 3.1.1.3
Factor out of .
Step 3.1.1.4
Factor out of .
Step 3.1.1.5
Factor out of .
Step 3.1.2
Factor.
Step 3.1.2.1
Factor using the AC method.
Step 3.1.2.1.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 3.1.2.1.2
Write the factored form using these integers.
Step 3.1.2.2
Remove unnecessary parentheses.
Step 3.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.3
Set equal to and solve for .
Step 3.3.1
Set equal to .
Step 3.3.2
Add to both sides of the equation.
Step 3.4
Set equal to and solve for .
Step 3.4.1
Set equal to .
Step 3.4.2
Subtract from both sides of the equation.
Step 3.5
The final solution is all the values that make true.
Step 4
Solve the original function at .
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Simplify each term.
Step 4.2.1.1
One to any power is one.
Step 4.2.1.2
Multiply by .
Step 4.2.1.3
One to any power is one.
Step 4.2.1.4
Multiply by .
Step 4.2.1.5
Multiply by .
Step 4.2.2
Step 4.2.2.1
Step 4.2.2.2
Subtract from .
Step 4.2.2.3
Step 4.2.3
Step 5
Solve the original function at .
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Simplify each term.
Step 5.2.1.1
Raise to the power of .
Step 5.2.1.2
Multiply by .
Step 5.2.1.3
Raise to the power of .
Step 5.2.1.4
Multiply by .
Step 5.2.1.5
Multiply by .
Step 5.2.2