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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 Power Rule which states that is where .
Step 1.3.3
To write as a fraction with a common denominator, multiply by .
Step 1.3.4
Combine and .
Step 1.3.5
Combine the numerators over the common denominator.
Step 1.3.6
Simplify the numerator.
Step 1.3.6.1
Multiply by .
Step 1.3.6.2
Subtract from .
Step 1.3.7
Move the negative in front of the fraction.
Step 1.3.8
Combine and .
Step 1.3.9
Combine and .
Step 1.3.10
Move to the denominator using the negative exponent rule .
Step 1.3.11
Factor out of .
Step 1.3.12
Cancel the common factors.
Step 1.3.12.1
Factor out of .
Step 1.3.12.2
Cancel the common factor.
Step 1.3.12.3
Rewrite the expression.
Step 1.3.13
Move the negative in front of the fraction.
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
Rewrite as .
Step 2.2.3
Differentiate using the chain rule, which states that is where and .
Step 2.2.3.1
To apply the Chain Rule, set as .
Step 2.2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3.3
Replace all occurrences of with .
Step 2.2.4
Differentiate using the Power Rule which states that is where .
Step 2.2.5
Multiply the exponents in .
Step 2.2.5.1
Apply the power rule and multiply exponents, .
Step 2.2.5.2
Multiply .
Step 2.2.5.2.1
Combine and .
Step 2.2.5.2.2
Multiply by .
Step 2.2.5.3
Move the negative in front of the fraction.
Step 2.2.6
To write as a fraction with a common denominator, multiply by .
Step 2.2.7
Combine and .
Step 2.2.8
Combine the numerators over the common denominator.
Step 2.2.9
Simplify the numerator.
Step 2.2.9.1
Multiply by .
Step 2.2.9.2
Subtract from .
Step 2.2.10
Move the negative in front of the fraction.
Step 2.2.11
Combine and .
Step 2.2.12
Combine and .
Step 2.2.13
Multiply by by adding the exponents.
Step 2.2.13.1
Move .
Step 2.2.13.2
Use the power rule to combine exponents.
Step 2.2.13.3
Combine the numerators over the common denominator.
Step 2.2.13.4
Subtract from .
Step 2.2.13.5
Move the negative in front of the fraction.
Step 2.2.14
Move to the denominator using the negative exponent rule .
Step 2.2.15
Multiply by .
Step 2.2.16
Combine and .
Step 2.2.17
Multiply by .
Step 2.2.18
Factor out of .
Step 2.2.19
Cancel the common factors.
Step 2.2.19.1
Factor out of .
Step 2.2.19.2
Cancel the common factor.
Step 2.2.19.3
Rewrite the expression.
Step 2.3
Add and .
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 Power Rule which states that is where .
Step 4.1.3.3
To write as a fraction with a common denominator, multiply by .
Step 4.1.3.4
Combine and .
Step 4.1.3.5
Combine the numerators over the common denominator.
Step 4.1.3.6
Simplify the numerator.
Step 4.1.3.6.1
Multiply by .
Step 4.1.3.6.2
Subtract from .
Step 4.1.3.7
Move the negative in front of the fraction.
Step 4.1.3.8
Combine and .
Step 4.1.3.9
Combine and .
Step 4.1.3.10
Move to the denominator using the negative exponent rule .
Step 4.1.3.11
Factor out of .
Step 4.1.3.12
Cancel the common factors.
Step 4.1.3.12.1
Factor out of .
Step 4.1.3.12.2
Cancel the common factor.
Step 4.1.3.12.3
Rewrite the expression.
Step 4.1.3.13
Move the negative in front of the fraction.
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
Find the LCD of the terms in the equation.
Step 5.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 5.3.2
The LCM of one and any expression is the expression.
Step 5.4
Multiply each term in by to eliminate the fractions.
Step 5.4.1
Multiply each term in by .
Step 5.4.2
Simplify the left side.
Step 5.4.2.1
Cancel the common factor of .
Step 5.4.2.1.1
Move the leading negative in into the numerator.
Step 5.4.2.1.2
Cancel the common factor.
Step 5.4.2.1.3
Rewrite the expression.
Step 5.5
Solve the equation.
Step 5.5.1
Rewrite the equation as .
Step 5.5.2
Divide each term in by and simplify.
Step 5.5.2.1
Divide each term in by .
Step 5.5.2.2
Simplify the left side.
Step 5.5.2.2.1
Cancel the common factor.
Step 5.5.2.2.2
Divide by .
Step 5.5.2.3
Simplify the right side.
Step 5.5.2.3.1
Divide by .
Step 5.5.3
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 5.5.4
Simplify the exponent.
Step 5.5.4.1
Simplify the left side.
Step 5.5.4.1.1
Simplify .
Step 5.5.4.1.1.1
Multiply the exponents in .
Step 5.5.4.1.1.1.1
Apply the power rule and multiply exponents, .
Step 5.5.4.1.1.1.2
Cancel the common factor of .
Step 5.5.4.1.1.1.2.1
Cancel the common factor.
Step 5.5.4.1.1.1.2.2
Rewrite the expression.
Step 5.5.4.1.1.1.3
Cancel the common factor of .
Step 5.5.4.1.1.1.3.1
Cancel the common factor.
Step 5.5.4.1.1.1.3.2
Rewrite the expression.
Step 5.5.4.1.1.2
Simplify.
Step 5.5.4.2
Simplify the right side.
Step 5.5.4.2.1
Simplify .
Step 5.5.4.2.1.1
Simplify the expression.
Step 5.5.4.2.1.1.1
Rewrite as .
Step 5.5.4.2.1.1.2
Apply the power rule and multiply exponents, .
Step 5.5.4.2.1.2
Cancel the common factor of .
Step 5.5.4.2.1.2.1
Cancel the common factor.
Step 5.5.4.2.1.2.2
Rewrite the expression.
Step 5.5.4.2.1.3
Raise to the power of .
Step 5.5.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.5.5.1
First, use the positive value of the to find the first solution.
Step 5.5.5.2
Next, use the negative value of the to find the second solution.
Step 5.5.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 6
Step 6.1
Apply the rule to rewrite the exponentiation as a radical.
Step 6.2
Set the denominator in equal to to find where the expression is undefined.
Step 6.3
Solve for .
Step 6.3.1
To remove the radical on the left side of the equation, cube both sides of the equation.
Step 6.3.2
Simplify each side of the equation.
Step 6.3.2.1
Use to rewrite as .
Step 6.3.2.2
Simplify the left side.
Step 6.3.2.2.1
Multiply the exponents in .
Step 6.3.2.2.1.1
Apply the power rule and multiply exponents, .
Step 6.3.2.2.1.2
Cancel the common factor of .
Step 6.3.2.2.1.2.1
Cancel the common factor.
Step 6.3.2.2.1.2.2
Rewrite the expression.
Step 6.3.2.3
Simplify the right side.
Step 6.3.2.3.1
Raising to any positive power yields .
Step 6.3.3
Solve for .
Step 6.3.3.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 6.3.3.2
Simplify .
Step 6.3.3.2.1
Rewrite as .
Step 6.3.3.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 6.3.3.2.3
Plus or minus is .
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
Move to the denominator using the negative exponent rule .
Step 9.2
Multiply by by adding the exponents.
Step 9.2.1
Use the power rule to combine exponents.
Step 9.2.2
To write as a fraction with a common denominator, multiply by .
Step 9.2.3
Combine and .
Step 9.2.4
Combine the numerators over the common denominator.
Step 9.2.5
Simplify the numerator.
Step 9.2.5.1
Multiply by .
Step 9.2.5.2
Subtract from .
Step 9.3
Simplify the denominator.
Step 9.3.1
Rewrite as .
Step 9.3.2
Apply the power rule and multiply exponents, .
Step 9.3.3
Cancel the common factor of .
Step 9.3.3.1
Cancel the common factor.
Step 9.3.3.2
Rewrite the expression.
Step 9.3.4
Raise to the power of .
Step 10
is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.
is a local minimum
Step 11
Step 11.1
Replace the variable with in the expression.
Step 11.2
Simplify the result.
Step 11.2.1
Simplify each term.
Step 11.2.1.1
Multiply by .
Step 11.2.1.2
Rewrite as .
Step 11.2.1.3
Apply the power rule and multiply exponents, .
Step 11.2.1.4
Cancel the common factor of .
Step 11.2.1.4.1
Cancel the common factor.
Step 11.2.1.4.2
Rewrite the expression.
Step 11.2.1.5
Evaluate the exponent.
Step 11.2.1.6
Multiply by .
Step 11.2.2
Subtract from .
Step 11.2.3
The final answer is .
Step 12
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 13
Step 13.1
Simplify the denominator.
Step 13.1.1
Rewrite as .
Step 13.1.2
Apply the power rule and multiply exponents, .
Step 13.1.3
Cancel the common factor of .
Step 13.1.3.1
Cancel the common factor.
Step 13.1.3.2
Rewrite the expression.
Step 13.1.4
Raise to the power of .
Step 13.2
Reduce the expression by cancelling the common factors.
Step 13.2.1
Cancel the common factor of and .
Step 13.2.1.1
Factor out of .
Step 13.2.1.2
Cancel the common factors.
Step 13.2.1.2.1
Factor out of .
Step 13.2.1.2.2
Cancel the common factor.
Step 13.2.1.2.3
Rewrite the expression.
Step 13.2.2
Move the negative in front of the fraction.
Step 14
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 15
Step 15.1
Replace the variable with in the expression.
Step 15.2
Simplify the result.
Step 15.2.1
Simplify each term.
Step 15.2.1.1
Multiply by .
Step 15.2.1.2
Rewrite as .
Step 15.2.1.3
Apply the power rule and multiply exponents, .
Step 15.2.1.4
Cancel the common factor of .
Step 15.2.1.4.1
Cancel the common factor.
Step 15.2.1.4.2
Rewrite the expression.
Step 15.2.1.5
Evaluate the exponent.
Step 15.2.1.6
Multiply by .
Step 15.2.2
Add and .
Step 15.2.3
The final answer is .
Step 16
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 17
Step 17.1
Simplify the expression.
Step 17.1.1
Rewrite as .
Step 17.1.2
Apply the power rule and multiply exponents, .
Step 17.2
Cancel the common factor of .
Step 17.2.1
Cancel the common factor.
Step 17.2.2
Rewrite the expression.
Step 17.3
Raising to any positive power yields .
Step 17.4
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Step 18
Step 18.1
Split into separate intervals around the values that make the first derivative or undefined.
Step 18.2
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Step 18.2.1
Replace the variable with in the expression.
Step 18.2.2
The final answer is .
Step 18.3
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Step 18.3.1
Replace the variable with in the expression.
Step 18.3.2
The final answer is .
Step 18.4
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Step 18.4.1
Replace the variable with in the expression.
Step 18.4.2
Simplify the result.
Step 18.4.2.1
Remove parentheses.
Step 18.4.2.2
The final answer is .
Step 18.5
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Step 18.5.1
Replace the variable with in the expression.
Step 18.5.2
Simplify the result.
Step 18.5.2.1
Remove parentheses.
Step 18.5.2.2
The final answer is .
Step 18.6
Since the first derivative changed signs from positive to negative around , then is a local maximum.
is a local maximum
Step 18.7
Since the first derivative did not change signs around , this is not a local maximum or minimum.
Not a local maximum or minimum
Step 18.8
Since the first derivative changed signs from negative to positive around , then is a local minimum.
is a local minimum
Step 18.9
These are the local extrema for .
is a local maximum
is a local minimum
is a local maximum
is a local minimum
Step 19