Calculus Examples

Find the Horizontal Tangent Line y=x^4-3x^2+2
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 2.3
Differentiate using the Constant Rule.
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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
Add and .
Step 3
Set the derivative equal to then solve the equation .
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Step 3.1
Factor out of .
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Step 3.1.1
Factor out of .
Step 3.1.2
Factor out of .
Step 3.1.3
Factor out of .
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 .
Step 3.4
Set equal to and solve for .
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Step 3.4.1
Set equal to .
Step 3.4.2
Solve for .
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Step 3.4.2.1
Add to both sides of the equation.
Step 3.4.2.2
Divide each term in by and simplify.
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Step 3.4.2.2.1
Divide each term in by .
Step 3.4.2.2.2
Simplify the left side.
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Step 3.4.2.2.2.1
Cancel the common factor of .
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Step 3.4.2.2.2.1.1
Cancel the common factor.
Step 3.4.2.2.2.1.2
Divide by .
Step 3.4.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3.4.2.4
Simplify .
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Step 3.4.2.4.1
Rewrite as .
Step 3.4.2.4.2
Multiply by .
Step 3.4.2.4.3
Combine and simplify the denominator.
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Step 3.4.2.4.3.1
Multiply by .
Step 3.4.2.4.3.2
Raise to the power of .
Step 3.4.2.4.3.3
Raise to the power of .
Step 3.4.2.4.3.4
Use the power rule to combine exponents.
Step 3.4.2.4.3.5
Add and .
Step 3.4.2.4.3.6
Rewrite as .
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Step 3.4.2.4.3.6.1
Use to rewrite as .
Step 3.4.2.4.3.6.2
Apply the power rule and multiply exponents, .
Step 3.4.2.4.3.6.3
Combine and .
Step 3.4.2.4.3.6.4
Cancel the common factor of .
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Step 3.4.2.4.3.6.4.1
Cancel the common factor.
Step 3.4.2.4.3.6.4.2
Rewrite the expression.
Step 3.4.2.4.3.6.5
Evaluate the exponent.
Step 3.4.2.4.4
Simplify the numerator.
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Step 3.4.2.4.4.1
Combine using the product rule for radicals.
Step 3.4.2.4.4.2
Multiply by .
Step 3.4.2.5
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.4.2.5.1
First, use the positive value of the to find the first solution.
Step 3.4.2.5.2
Next, use the negative value of the to find the second solution.
Step 3.4.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
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
Raising to any positive power yields .
Step 4.2.1.2
Raising to any positive power yields .
Step 4.2.1.3
Multiply by .
Step 4.2.2
Simplify by adding numbers.
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Step 4.2.2.1
Add and .
Step 4.2.2.2
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
Apply the product rule to .
Step 5.2.1.2
Simplify the numerator.
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Step 5.2.1.2.1
Rewrite as .
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Step 5.2.1.2.1.1
Use to rewrite as .
Step 5.2.1.2.1.2
Apply the power rule and multiply exponents, .
Step 5.2.1.2.1.3
Combine and .
Step 5.2.1.2.1.4
Cancel the common factor of and .
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Step 5.2.1.2.1.4.1
Factor out of .
Step 5.2.1.2.1.4.2
Cancel the common factors.
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Step 5.2.1.2.1.4.2.1
Factor out of .
Step 5.2.1.2.1.4.2.2
Cancel the common factor.
Step 5.2.1.2.1.4.2.3
Rewrite the expression.
Step 5.2.1.2.1.4.2.4
Divide by .
Step 5.2.1.2.2
Raise to the power of .
Step 5.2.1.3
Raise to the power of .
Step 5.2.1.4
Cancel the common factor of and .
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Step 5.2.1.4.1
Factor out of .
Step 5.2.1.4.2
Cancel the common factors.
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Step 5.2.1.4.2.1
Factor out of .
Step 5.2.1.4.2.2
Cancel the common factor.
Step 5.2.1.4.2.3
Rewrite the expression.
Step 5.2.1.5
Apply the product rule to .
Step 5.2.1.6
Rewrite as .
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Step 5.2.1.6.1
Use to rewrite as .
Step 5.2.1.6.2
Apply the power rule and multiply exponents, .
Step 5.2.1.6.3
Combine and .
Step 5.2.1.6.4
Cancel the common factor of .
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Step 5.2.1.6.4.1
Cancel the common factor.
Step 5.2.1.6.4.2
Rewrite the expression.
Step 5.2.1.6.5
Evaluate the exponent.
Step 5.2.1.7
Raise to the power of .
Step 5.2.1.8
Cancel the common factor of and .
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Step 5.2.1.8.1
Factor out of .
Step 5.2.1.8.2
Cancel the common factors.
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Step 5.2.1.8.2.1
Factor out of .
Step 5.2.1.8.2.2
Cancel the common factor.
Step 5.2.1.8.2.3
Rewrite the expression.
Step 5.2.1.9
Multiply .
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Step 5.2.1.9.1
Combine and .
Step 5.2.1.9.2
Multiply by .
Step 5.2.1.10
Move the negative in front of the fraction.
Step 5.2.2
Find the common denominator.
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Step 5.2.2.1
Multiply by .
Step 5.2.2.2
Multiply by .
Step 5.2.2.3
Write as a fraction with denominator .
Step 5.2.2.4
Multiply by .
Step 5.2.2.5
Multiply by .
Step 5.2.2.6
Multiply by .
Step 5.2.3
Combine the numerators over the common denominator.
Step 5.2.4
Simplify each term.
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Step 5.2.4.1
Multiply by .
Step 5.2.4.2
Multiply by .
Step 5.2.5
Simplify the expression.
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Step 5.2.5.1
Subtract from .
Step 5.2.5.2
Add and .
Step 5.2.5.3
Move the negative in front of the fraction.
Step 5.2.6
The final answer is .
Step 6
The horizontal tangent lines on function are .
Step 7