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Calculus Examples
Step 1
Write as a function.
Step 2
Differentiate using the function rule which states that the derivative of is .
Step 3
Step 3.1
Cancel the common factor of .
Step 3.1.1
Cancel the common factor.
Step 3.1.2
Rewrite the expression.
Step 3.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.3
Rewrite as .
Step 3.4
Differentiate using the Power Rule which states that is where .
Step 3.5
Simplify.
Step 3.5.1
Rewrite the expression using the negative exponent rule .
Step 3.5.2
Combine and .
Step 4
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 5
Step 5.1
Differentiate using the function rule which states that the derivative of is .
Step 5.2
The first derivative of with respect to is .
Step 6
Step 6.1
Set the first derivative equal to .
Step 6.2
Cancel the common factor of .
Step 6.2.1
Cancel the common factor.
Step 6.2.2
Rewrite the expression.
Step 6.3
Multiply both sides of the equation by .
Step 6.4
Rewrite the equation as .
Step 6.5
Multiply by .
Step 6.6
Rewrite so is on the left side.
Step 6.7
The variable got canceled.
All real numbers
All real numbers
Step 7
Step 7.1
Set the denominator in equal to to find where the expression is undefined.
Step 7.2
Divide each term in by and simplify.
Step 7.2.1
Divide each term in by .
Step 7.2.2
Simplify the left side.
Step 7.2.2.1
Cancel the common factor of .
Step 7.2.2.1.1
Cancel the common factor.
Step 7.2.2.1.2
Divide by .
Step 7.2.3
Simplify the right side.
Step 7.2.3.1
Divide by .
Step 8
Critical points to evaluate.
real
Step 9
Evaluate the second derivative at . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.
Step 10
Step 10.1
Combine exponents.
Step 10.1.1
Raise to the power of .
Step 10.1.2
Raise to the power of .
Step 10.1.3
Use the power rule to combine exponents.
Step 10.1.4
Add and .
Step 10.1.5
Raise to the power of .
Step 10.1.6
Use the power rule to combine exponents.
Step 10.1.7
Add and .
Step 10.1.8
Raise to the power of .
Step 10.1.9
Raise to the power of .
Step 10.1.10
Use the power rule to combine exponents.
Step 10.1.11
Add and .
Step 10.1.12
Raise to the power of .
Step 10.1.13
Raise to the power of .
Step 10.1.14
Use the power rule to combine exponents.
Step 10.1.15
Add and .
Step 10.2
Apply the product rule to .
Step 10.3
Use the power rule to distribute the exponent.
Step 10.3.1
Apply the product rule to .
Step 10.3.2
Apply the product rule to .
Step 10.3.3
Apply the product rule to .
Step 10.3.4
Apply the product rule to .
Step 10.3.5
Apply the product rule to .
Step 10.3.6
Apply the product rule to .
Step 10.3.7
Apply the product rule to .
Step 10.3.8
Apply the product rule to .
Step 10.4
Multiply the exponents in .
Step 10.4.1
Apply the power rule and multiply exponents, .
Step 10.4.2
Multiply by .
Step 10.5
Multiply the exponents in .
Step 10.5.1
Apply the power rule and multiply exponents, .
Step 10.5.2
Multiply by .
Step 10.6
Multiply the exponents in .
Step 10.6.1
Apply the power rule and multiply exponents, .
Step 10.6.2
Multiply by .
Step 11
Since the first derivative test failed, there are no local extrema.
No Local Extrema
Step 12