Calculus Examples

Find the Critical Points f(x)=(x^2)/(x+1)
Step 1
Find the first derivative.
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Step 1.1
Find the first derivative.
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Step 1.1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.2
Differentiate.
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Step 1.1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2
Move to the left of .
Step 1.1.2.3
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.4
Differentiate using the Power Rule which states that is where .
Step 1.1.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.6
Simplify the expression.
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Step 1.1.2.6.1
Add and .
Step 1.1.2.6.2
Multiply by .
Step 1.1.3
Simplify.
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Step 1.1.3.1
Apply the distributive property.
Step 1.1.3.2
Apply the distributive property.
Step 1.1.3.3
Simplify the numerator.
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Step 1.1.3.3.1
Simplify each term.
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Step 1.1.3.3.1.1
Multiply by by adding the exponents.
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Step 1.1.3.3.1.1.1
Move .
Step 1.1.3.3.1.1.2
Multiply by .
Step 1.1.3.3.1.2
Multiply by .
Step 1.1.3.3.2
Subtract from .
Step 1.1.3.4
Factor out of .
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Step 1.1.3.4.1
Factor out of .
Step 1.1.3.4.2
Factor out of .
Step 1.1.3.4.3
Factor out of .
Step 1.2
The first derivative of with respect to is .
Step 2
Set the first derivative equal to then solve the equation .
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Step 2.1
Set the first derivative equal to .
Step 2.2
Set the numerator equal to zero.
Step 2.3
Solve the equation for .
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Step 2.3.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.3.2
Set equal to .
Step 2.3.3
Set equal to and solve for .
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Step 2.3.3.1
Set equal to .
Step 2.3.3.2
Subtract from both sides of the equation.
Step 2.3.4
The final solution is all the values that make true.
Step 3
Find the values where the derivative is undefined.
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Step 3.1
Set the denominator in equal to to find where the expression is undefined.
Step 3.2
Solve for .
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Step 3.2.1
Set the equal to .
Step 3.2.2
Subtract from both sides of the equation.
Step 4
Evaluate at each value where the derivative is or undefined.
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Step 4.1
Evaluate at .
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Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
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Step 4.1.2.1
Raising to any positive power yields .
Step 4.1.2.2
Add and .
Step 4.1.2.3
Divide by .
Step 4.2
Evaluate at .
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Step 4.2.1
Substitute for .
Step 4.2.2
Simplify.
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Step 4.2.2.1
Raise to the power of .
Step 4.2.2.2
Add and .
Step 4.2.2.3
Divide by .
Step 4.3
Evaluate at .
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Step 4.3.1
Substitute for .
Step 4.3.2
Simplify.
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Step 4.3.2.1
Add and .
Step 4.3.2.2
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Undefined
Step 4.4
List all of the points.
Step 5