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Calculus Examples
Step 1
Step 1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2
Evaluate .
Step 1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.2.3
Multiply by .
Step 1.3
Evaluate .
Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Product Rule which states that is where and .
Step 1.3.3
The derivative of with respect to is .
Step 1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.3.5
Combine and .
Step 1.3.6
Cancel the common factor of .
Step 1.3.6.1
Cancel the common factor.
Step 1.3.6.2
Rewrite the expression.
Step 1.3.7
Multiply by .
Step 1.4
Simplify.
Step 1.4.1
Apply the distributive property.
Step 1.4.2
Combine terms.
Step 1.4.2.1
Multiply by .
Step 1.4.2.2
Subtract from .
Step 2
Step 2.1
Differentiate.
Step 2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Evaluate .
Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
The derivative of with respect to is .
Step 2.2.3
Combine and .
Step 2.2.4
Move the negative in front of the fraction.
Step 2.3
Subtract from .
Step 3
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 4
Step 4.1
Find the first derivative.
Step 4.1.1
By the Sum Rule, the derivative of with respect to is .
Step 4.1.2
Evaluate .
Step 4.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.2.2
Differentiate using the Power Rule which states that is where .
Step 4.1.2.3
Multiply by .
Step 4.1.3
Evaluate .
Step 4.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.3.2
Differentiate using the Product Rule which states that is where and .
Step 4.1.3.3
The derivative of with respect to is .
Step 4.1.3.4
Differentiate using the Power Rule which states that is where .
Step 4.1.3.5
Combine and .
Step 4.1.3.6
Cancel the common factor of .
Step 4.1.3.6.1
Cancel the common factor.
Step 4.1.3.6.2
Rewrite the expression.
Step 4.1.3.7
Multiply by .
Step 4.1.4
Simplify.
Step 4.1.4.1
Apply the distributive property.
Step 4.1.4.2
Combine terms.
Step 4.1.4.2.1
Multiply by .
Step 4.1.4.2.2
Subtract from .
Step 4.2
The first derivative of with respect to is .
Step 5
Step 5.1
Set the first derivative equal to .
Step 5.2
Subtract from both sides of the equation.
Step 5.3
Divide each term in by and simplify.
Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
Step 5.3.2.1
Cancel the common factor of .
Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Divide by .
Step 5.3.3
Simplify the right side.
Step 5.3.3.1
Dividing two negative values results in a positive value.
Step 5.4
To solve for , rewrite the equation using properties of logarithms.
Step 5.5
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 5.6
Rewrite the equation as .
Step 6
Step 6.1
Set the argument in less than or equal to to find where the expression is undefined.
Step 6.2
The equation is undefined where the denominator equals , the argument of a square root is less than , or the argument of a logarithm is less than or equal to .
Step 7
Critical points to evaluate.
Step 8
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 9
is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.
is a local maximum
Step 10
Step 10.1
Replace the variable with in the expression.
Step 10.2
Simplify the result.
Step 10.2.1
Simplify each term.
Step 10.2.1.1
Use logarithm rules to move out of the exponent.
Step 10.2.1.2
The natural logarithm of is .
Step 10.2.1.3
Multiply by .
Step 10.2.1.4
Cancel the common factor of .
Step 10.2.1.4.1
Factor out of .
Step 10.2.1.4.2
Cancel the common factor.
Step 10.2.1.4.3
Rewrite the expression.
Step 10.2.1.5
Multiply by .
Step 10.2.2
Subtract from .
Step 10.2.3
The final answer is .
Step 11
These are the local extrema for .
is a local maxima
Step 12