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.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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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 .
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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 .
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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.
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Step 2.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.2
Add and .
Step 3
Set the derivative equal to then solve the equation .
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Step 3.1
Factor the left side of the equation.
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Step 3.1.1
Factor out of .
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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.
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Step 3.1.2.1
Factor using the AC method.
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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 .
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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 .
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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 .
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Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
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Step 4.2.1
Simplify each term.
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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
Simplify by adding and subtracting.
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Step 4.2.2.1
Add and .
Step 4.2.2.2
Subtract from .
Step 4.2.2.3
Add and .
Step 4.2.3
The final answer is .
Step 5
Solve the original function at .
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Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
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Step 5.2.1
Simplify each term.
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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
Simplify by adding numbers.
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Step 5.2.2.1
Add and .
Step 5.2.2.2
Add and .
Step 5.2.2.3
Add and .
Step 5.2.3
The final answer is .
Step 6
The horizontal tangent lines on function are .
Step 7
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