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Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
By the Sum Rule, 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
Differentiate using the Power Rule which states that is where .
Step 2.2.3
To write as a fraction with a common denominator, multiply by .
Step 2.2.4
Combine and .
Step 2.2.5
Combine the numerators over the common denominator.
Step 2.2.6
Simplify the numerator.
Step 2.2.6.1
Multiply by .
Step 2.2.6.2
Subtract from .
Step 2.2.7
Move the negative in front of the fraction.
Step 2.2.8
Combine and .
Step 2.2.9
Combine and .
Step 2.2.10
Multiply by .
Step 2.2.11
Move to the denominator using the negative exponent rule .
Step 2.2.12
Factor out of .
Step 2.2.13
Cancel the common factors.
Step 2.2.13.1
Factor out of .
Step 2.2.13.2
Cancel the common factor.
Step 2.2.13.3
Rewrite the expression.
Step 2.3
Evaluate .
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 3
Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Rewrite as .
Step 3.2.3
Differentiate using the chain rule, which states that is where and .
Step 3.2.3.1
To apply the Chain Rule, set as .
Step 3.2.3.2
Differentiate using the Power Rule which states that is where .
Step 3.2.3.3
Replace all occurrences of with .
Step 3.2.4
Differentiate using the Power Rule which states that is where .
Step 3.2.5
Multiply the exponents in .
Step 3.2.5.1
Apply the power rule and multiply exponents, .
Step 3.2.5.2
Combine and .
Step 3.2.5.3
Move the negative in front of the fraction.
Step 3.2.6
To write as a fraction with a common denominator, multiply by .
Step 3.2.7
Combine and .
Step 3.2.8
Combine the numerators over the common denominator.
Step 3.2.9
Simplify the numerator.
Step 3.2.9.1
Multiply by .
Step 3.2.9.2
Subtract from .
Step 3.2.10
Move the negative in front of the fraction.
Step 3.2.11
Combine and .
Step 3.2.12
Combine and .
Step 3.2.13
Multiply by by adding the exponents.
Step 3.2.13.1
Use the power rule to combine exponents.
Step 3.2.13.2
Combine the numerators over the common denominator.
Step 3.2.13.3
Subtract from .
Step 3.2.13.4
Move the negative in front of the fraction.
Step 3.2.14
Move to the denominator using the negative exponent rule .
Step 3.2.15
Multiply by .
Step 3.2.16
Combine and .
Step 3.2.17
Move the negative in front of the fraction.
Step 3.3
Differentiate using the Constant Rule.
Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Add 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
Find the first derivative.
Step 5.1.1
By the Sum Rule, the derivative of with respect to is .
Step 5.1.2
Evaluate .
Step 5.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.2.2
Differentiate using the Power Rule which states that is where .
Step 5.1.2.3
To write as a fraction with a common denominator, multiply by .
Step 5.1.2.4
Combine and .
Step 5.1.2.5
Combine the numerators over the common denominator.
Step 5.1.2.6
Simplify the numerator.
Step 5.1.2.6.1
Multiply by .
Step 5.1.2.6.2
Subtract from .
Step 5.1.2.7
Move the negative in front of the fraction.
Step 5.1.2.8
Combine and .
Step 5.1.2.9
Combine and .
Step 5.1.2.10
Multiply by .
Step 5.1.2.11
Move to the denominator using the negative exponent rule .
Step 5.1.2.12
Factor out of .
Step 5.1.2.13
Cancel the common factors.
Step 5.1.2.13.1
Factor out of .
Step 5.1.2.13.2
Cancel the common factor.
Step 5.1.2.13.3
Rewrite the expression.
Step 5.1.3
Evaluate .
Step 5.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3.2
Differentiate using the Power Rule which states that is where .
Step 5.1.3.3
Multiply by .
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
Add to both sides of the equation.
Step 6.3
Find the LCD of the terms in the equation.
Step 6.3.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 6.3.2
The LCM of one and any expression is the expression.
Step 6.4
Multiply each term in by to eliminate the fractions.
Step 6.4.1
Multiply each term in by .
Step 6.4.2
Simplify the left side.
Step 6.4.2.1
Cancel the common factor of .
Step 6.4.2.1.1
Cancel the common factor.
Step 6.4.2.1.2
Rewrite the expression.
Step 6.5
Solve the equation.
Step 6.5.1
Rewrite the equation as .
Step 6.5.2
Divide each term in by and simplify.
Step 6.5.2.1
Divide each term in by .
Step 6.5.2.2
Simplify the left side.
Step 6.5.2.2.1
Cancel the common factor.
Step 6.5.2.2.2
Divide by .
Step 6.5.2.3
Simplify the right side.
Step 6.5.2.3.1
Divide by .
Step 6.5.3
Raise each side of the equation to the power of to eliminate the fractional exponent on the left side.
Step 6.5.4
Simplify the exponent.
Step 6.5.4.1
Simplify the left side.
Step 6.5.4.1.1
Simplify .
Step 6.5.4.1.1.1
Multiply the exponents in .
Step 6.5.4.1.1.1.1
Apply the power rule and multiply exponents, .
Step 6.5.4.1.1.1.2
Cancel the common factor of .
Step 6.5.4.1.1.1.2.1
Cancel the common factor.
Step 6.5.4.1.1.1.2.2
Rewrite the expression.
Step 6.5.4.1.1.2
Simplify.
Step 6.5.4.2
Simplify the right side.
Step 6.5.4.2.1
One to any power is one.
Step 7
Step 7.1
Convert expressions with fractional exponents to radicals.
Step 7.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 7.1.2
Anything raised to is the base itself.
Step 7.2
Set the denominator in equal to to find where the expression is undefined.
Step 7.3
Solve for .
Step 7.3.1
To remove the radical on the left side of the equation, cube both sides of the equation.
Step 7.3.2
Simplify each side of the equation.
Step 7.3.2.1
Use to rewrite as .
Step 7.3.2.2
Simplify the left side.
Step 7.3.2.2.1
Simplify .
Step 7.3.2.2.1.1
Multiply the exponents in .
Step 7.3.2.2.1.1.1
Apply the power rule and multiply exponents, .
Step 7.3.2.2.1.1.2
Cancel the common factor of .
Step 7.3.2.2.1.1.2.1
Cancel the common factor.
Step 7.3.2.2.1.1.2.2
Rewrite the expression.
Step 7.3.2.2.1.2
Simplify.
Step 7.3.2.3
Simplify the right side.
Step 7.3.2.3.1
Raising to any positive power yields .
Step 8
Critical points to evaluate.
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
One to any power is one.
Step 10.2
Multiply by .
Step 11
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 12
Step 12.1
Replace the variable with in the expression.
Step 12.2
Simplify the result.
Step 12.2.1
Simplify each term.
Step 12.2.1.1
One to any power is one.
Step 12.2.1.2
Multiply by .
Step 12.2.1.3
Multiply by .
Step 12.2.2
Subtract from .
Step 12.2.3
The final answer is .
Step 13
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 14
Step 14.1
Simplify the expression.
Step 14.1.1
Rewrite as .
Step 14.1.2
Apply the power rule and multiply exponents, .
Step 14.2
Cancel the common factor of .
Step 14.2.1
Cancel the common factor.
Step 14.2.2
Rewrite the expression.
Step 14.3
Simplify the expression.
Step 14.3.1
Raising to any positive power yields .
Step 14.3.2
Multiply by .
Step 14.3.3
The expression contains a division by . The expression is undefined.
Undefined
Step 14.4
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Step 15
Step 15.1
Split into separate intervals around the values that make the first derivative or undefined.
Step 15.2
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Step 15.2.1
Replace the variable with in the expression.
Step 15.2.2
The final answer is .
Step 15.3
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Step 15.3.1
Replace the variable with in the expression.
Step 15.3.2
Simplify the result.
Step 15.3.2.1
Simplify each term.
Step 15.3.2.1.1
Raise to the power of .
Step 15.3.2.1.2
Divide by .
Step 15.3.2.2
Subtract from .
Step 15.3.2.3
The final answer is .
Step 15.4
Substitute any number, such as , from the interval in the first derivative to check if the result is negative or positive.
Step 15.4.1
Replace the variable with in the expression.
Step 15.4.2
Simplify the result.
Step 15.4.2.1
Remove parentheses.
Step 15.4.2.2
The final answer is .
Step 15.5
Since the first derivative changed signs from negative to positive around , then is a local minimum.
is a local minimum
Step 15.6
Since the first derivative changed signs from positive to negative around , then is a local maximum.
is a local maximum
Step 15.7
These are the local extrema for .
is a local minimum
is a local maximum
is a local minimum
is a local maximum
Step 16