Calculus Examples

Find the Horizontal Tangent Line y=x^3+9x
Step 1
Set as a function of .
Step 2
Find the derivative.
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Step 2.1
Differentiate.
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Step 2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2
Differentiate using the Power Rule which states that is where .
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 3
Set the derivative equal to then solve the equation .
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Step 3.1
Subtract from both sides of the equation.
Step 3.2
Divide each term in by and simplify.
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Step 3.2.1
Divide each term in by .
Step 3.2.2
Simplify the left side.
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Step 3.2.2.1
Cancel the common factor of .
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Step 3.2.2.1.1
Cancel the common factor.
Step 3.2.2.1.2
Divide by .
Step 3.2.3
Simplify the right side.
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Step 3.2.3.1
Divide by .
Step 3.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.4
Simplify .
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Step 3.4.1
Rewrite as .
Step 3.4.2
Rewrite as .
Step 3.4.3
Rewrite as .
Step 3.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.5.1
First, use the positive value of the to find the first solution.
Step 3.5.2
Next, use the negative value of the to find the second solution.
Step 3.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
A tangent line cannot be found at an imaginary point. The point at does not exist on the real coordinate system.
A tangent cannot be found from the root
Step 5
There are no horizontal tangent lines on the function .
No horizontal tangent lines
Step 6