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Calculus Examples
Step 1
Step 1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.2
The derivative of with respect to is .
Step 1.3
Differentiate using the Power Rule.
Step 1.3.1
Combine and .
Step 1.3.2
Cancel the common factor of .
Step 1.3.2.1
Cancel the common factor.
Step 1.3.2.2
Rewrite the expression.
Step 1.3.3
Differentiate using the Power Rule which states that is where .
Step 1.3.4
Multiply by .
Step 2
Step 2.1
Differentiate using the Quotient Rule which states that is where and .
Step 2.2
Differentiate.
Step 2.2.1
Multiply the exponents in .
Step 2.2.1.1
Apply the power rule and multiply exponents, .
Step 2.2.1.2
Multiply by .
Step 2.2.2
By the Sum Rule, the derivative of with respect to is .
Step 2.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.4
Add and .
Step 2.2.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
The derivative of with respect to is .
Step 2.4
Differentiate using the Power Rule.
Step 2.4.1
Combine and .
Step 2.4.2
Cancel the common factor of and .
Step 2.4.2.1
Factor out of .
Step 2.4.2.2
Cancel the common factors.
Step 2.4.2.2.1
Raise to the power of .
Step 2.4.2.2.2
Factor out of .
Step 2.4.2.2.3
Cancel the common factor.
Step 2.4.2.2.4
Rewrite the expression.
Step 2.4.2.2.5
Divide by .
Step 2.4.3
Differentiate using the Power Rule which states that is where .
Step 2.4.4
Simplify with factoring out.
Step 2.4.4.1
Multiply by .
Step 2.4.4.2
Factor out of .
Step 2.4.4.2.1
Factor out of .
Step 2.4.4.2.2
Factor out of .
Step 2.4.4.2.3
Factor out of .
Step 2.5
Cancel the common factors.
Step 2.5.1
Factor out of .
Step 2.5.2
Cancel the common factor.
Step 2.5.3
Rewrite the expression.
Step 2.6
Simplify.
Step 2.6.1
Apply the distributive property.
Step 2.6.2
Simplify the numerator.
Step 2.6.2.1
Simplify each term.
Step 2.6.2.1.1
Multiply by .
Step 2.6.2.1.2
Multiply .
Step 2.6.2.1.2.1
Multiply by .
Step 2.6.2.1.2.2
Simplify by moving inside the logarithm.
Step 2.6.2.2
Subtract from .
Step 2.6.3
Rewrite as .
Step 2.6.4
Factor out of .
Step 2.6.5
Factor out of .
Step 2.6.6
Move the negative in front of the fraction.
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
Differentiate using the Quotient Rule which states that is where and .
Step 4.1.2
The derivative of with respect to is .
Step 4.1.3
Differentiate using the Power Rule.
Step 4.1.3.1
Combine and .
Step 4.1.3.2
Cancel the common factor of .
Step 4.1.3.2.1
Cancel the common factor.
Step 4.1.3.2.2
Rewrite the expression.
Step 4.1.3.3
Differentiate using the Power Rule which states that is where .
Step 4.1.3.4
Multiply by .
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
Set the numerator equal to zero.
Step 5.3
Solve the equation for .
Step 5.3.1
Subtract from both sides of the equation.
Step 5.3.2
Divide each term in by and simplify.
Step 5.3.2.1
Divide each term in by .
Step 5.3.2.2
Simplify the left side.
Step 5.3.2.2.1
Dividing two negative values results in a positive value.
Step 5.3.2.2.2
Divide by .
Step 5.3.2.3
Simplify the right side.
Step 5.3.2.3.1
Divide by .
Step 5.3.3
To solve for , rewrite the equation using properties of logarithms.
Step 5.3.4
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 5.3.5
Rewrite the equation as .
Step 6
Step 6.1
Set the denominator in equal to to find where the expression is undefined.
Step 6.2
Solve for .
Step 6.2.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 6.2.2
Simplify .
Step 6.2.2.1
Rewrite as .
Step 6.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 6.2.2.3
Plus or minus is .
Step 6.3
Set the argument in less than or equal to to find where the expression is undefined.
Step 6.4
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
Step 9.1
Use logarithm rules to move out of the exponent.
Step 9.2
The natural logarithm of is .
Step 9.3
Multiply by .
Step 9.4
Multiply by .
Step 9.5
Subtract from .
Step 10
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 11
Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
Step 11.2.1
The natural logarithm of is .
Step 11.2.2
The final answer is .
Step 12
These are the local extrema for .
is a local maxima
Step 13