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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
Multiply by .
Step 1.4
Differentiate using the Constant Rule.
Step 1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.4.2
Add and .
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
Multiply by .
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
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
Multiply by .
Step 4.1.4
Differentiate using the Constant Rule.
Step 4.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.4.2
Add and .
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
Factor out of .
Step 5.2.1
Factor out of .
Step 5.2.2
Factor out of .
Step 5.2.3
Factor out of .
Step 5.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5.4
Set equal to .
Step 5.5
Set equal to and solve for .
Step 5.5.1
Set equal to .
Step 5.5.2
Solve for .
Step 5.5.2.1
Add to both sides of the equation.
Step 5.5.2.2
Divide each term in by and simplify.
Step 5.5.2.2.1
Divide each term in by .
Step 5.5.2.2.2
Simplify the left side.
Step 5.5.2.2.2.1
Cancel the common factor of .
Step 5.5.2.2.2.1.1
Cancel the common factor.
Step 5.5.2.2.2.1.2
Divide by .
Step 5.5.2.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.5.2.4
Simplify .
Step 5.5.2.4.1
Rewrite as .
Step 5.5.2.4.2
Simplify the numerator.
Step 5.5.2.4.2.1
Rewrite as .
Step 5.5.2.4.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 5.5.2.4.3
Multiply by .
Step 5.5.2.4.4
Combine and simplify the denominator.
Step 5.5.2.4.4.1
Multiply by .
Step 5.5.2.4.4.2
Raise to the power of .
Step 5.5.2.4.4.3
Raise to the power of .
Step 5.5.2.4.4.4
Use the power rule to combine exponents.
Step 5.5.2.4.4.5
Add and .
Step 5.5.2.4.4.6
Rewrite as .
Step 5.5.2.4.4.6.1
Use to rewrite as .
Step 5.5.2.4.4.6.2
Apply the power rule and multiply exponents, .
Step 5.5.2.4.4.6.3
Combine and .
Step 5.5.2.4.4.6.4
Cancel the common factor of .
Step 5.5.2.4.4.6.4.1
Cancel the common factor.
Step 5.5.2.4.4.6.4.2
Rewrite the expression.
Step 5.5.2.4.4.6.5
Evaluate the exponent.
Step 5.5.2.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.5.2.5.1
First, use the positive value of the to find the first solution.
Step 5.5.2.5.2
Next, use the negative value of the to find the second solution.
Step 5.5.2.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.6
The final solution is all the values that make true.
Step 6
Step 6.1
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
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
Simplify each term.
Step 9.1.1
Raising to any positive power yields .
Step 9.1.2
Multiply by .
Step 9.2
Add and .
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
Raising to any positive power yields .
Step 11.2.1.2
Multiply by .
Step 11.2.1.3
Raising to any positive power yields .
Step 11.2.1.4
Multiply by .
Step 11.2.2
Simplify by adding and subtracting.
Step 11.2.2.1
Add and .
Step 11.2.2.2
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 each term.
Step 13.1.1
Use the power rule to distribute the exponent.
Step 13.1.1.1
Apply the product rule to .
Step 13.1.1.2
Apply the product rule to .
Step 13.1.2
Simplify the numerator.
Step 13.1.2.1
Raise to the power of .
Step 13.1.2.2
Rewrite as .
Step 13.1.2.2.1
Use to rewrite as .
Step 13.1.2.2.2
Apply the power rule and multiply exponents, .
Step 13.1.2.2.3
Combine and .
Step 13.1.2.2.4
Cancel the common factor of .
Step 13.1.2.2.4.1
Cancel the common factor.
Step 13.1.2.2.4.2
Rewrite the expression.
Step 13.1.2.2.5
Evaluate the exponent.
Step 13.1.3
Raise to the power of .
Step 13.1.4
Multiply by .
Step 13.1.5
Cancel the common factor of .
Step 13.1.5.1
Factor out of .
Step 13.1.5.2
Cancel the common factor.
Step 13.1.5.3
Rewrite the expression.
Step 13.1.6
Multiply by .
Step 13.2
Add and .
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
Use the power rule to distribute the exponent.
Step 15.2.1.1.1
Apply the product rule to .
Step 15.2.1.1.2
Apply the product rule to .
Step 15.2.1.2
Simplify the numerator.
Step 15.2.1.2.1
Raise to the power of .
Step 15.2.1.2.2
Rewrite as .
Step 15.2.1.2.2.1
Use to rewrite as .
Step 15.2.1.2.2.2
Apply the power rule and multiply exponents, .
Step 15.2.1.2.2.3
Combine and .
Step 15.2.1.2.2.4
Cancel the common factor of and .
Step 15.2.1.2.2.4.1
Factor out of .
Step 15.2.1.2.2.4.2
Cancel the common factors.
Step 15.2.1.2.2.4.2.1
Factor out of .
Step 15.2.1.2.2.4.2.2
Cancel the common factor.
Step 15.2.1.2.2.4.2.3
Rewrite the expression.
Step 15.2.1.2.2.4.2.4
Divide by .
Step 15.2.1.2.3
Raise to the power of .
Step 15.2.1.3
Raise to the power of .
Step 15.2.1.4
Multiply by .
Step 15.2.1.5
Cancel the common factor of and .
Step 15.2.1.5.1
Factor out of .
Step 15.2.1.5.2
Cancel the common factors.
Step 15.2.1.5.2.1
Factor out of .
Step 15.2.1.5.2.2
Cancel the common factor.
Step 15.2.1.5.2.3
Rewrite the expression.
Step 15.2.1.6
Use the power rule to distribute the exponent.
Step 15.2.1.6.1
Apply the product rule to .
Step 15.2.1.6.2
Apply the product rule to .
Step 15.2.1.7
Simplify the numerator.
Step 15.2.1.7.1
Raise to the power of .
Step 15.2.1.7.2
Rewrite as .
Step 15.2.1.7.2.1
Use to rewrite as .
Step 15.2.1.7.2.2
Apply the power rule and multiply exponents, .
Step 15.2.1.7.2.3
Combine and .
Step 15.2.1.7.2.4
Cancel the common factor of .
Step 15.2.1.7.2.4.1
Cancel the common factor.
Step 15.2.1.7.2.4.2
Rewrite the expression.
Step 15.2.1.7.2.5
Evaluate the exponent.
Step 15.2.1.8
Raise to the power of .
Step 15.2.1.9
Multiply by .
Step 15.2.1.10
Cancel the common factor of and .
Step 15.2.1.10.1
Factor out of .
Step 15.2.1.10.2
Cancel the common factors.
Step 15.2.1.10.2.1
Factor out of .
Step 15.2.1.10.2.2
Cancel the common factor.
Step 15.2.1.10.2.3
Rewrite the expression.
Step 15.2.1.11
Multiply .
Step 15.2.1.11.1
Combine and .
Step 15.2.1.11.2
Multiply by .
Step 15.2.2
Find the common denominator.
Step 15.2.2.1
Multiply by .
Step 15.2.2.2
Multiply by .
Step 15.2.2.3
Write as a fraction with denominator .
Step 15.2.2.4
Multiply by .
Step 15.2.2.5
Multiply by .
Step 15.2.2.6
Multiply by .
Step 15.2.3
Combine the numerators over the common denominator.
Step 15.2.4
Simplify each term.
Step 15.2.4.1
Multiply by .
Step 15.2.4.2
Multiply by .
Step 15.2.5
Simplify by adding and subtracting.
Step 15.2.5.1
Add and .
Step 15.2.5.2
Subtract from .
Step 15.2.6
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 each term.
Step 17.1.1
Use the power rule to distribute the exponent.
Step 17.1.1.1
Apply the product rule to .
Step 17.1.1.2
Apply the product rule to .
Step 17.1.1.3
Apply the product rule to .
Step 17.1.2
Raise to the power of .
Step 17.1.3
Multiply by .
Step 17.1.4
Simplify the numerator.
Step 17.1.4.1
Raise to the power of .
Step 17.1.4.2
Rewrite as .
Step 17.1.4.2.1
Use to rewrite as .
Step 17.1.4.2.2
Apply the power rule and multiply exponents, .
Step 17.1.4.2.3
Combine and .
Step 17.1.4.2.4
Cancel the common factor of .
Step 17.1.4.2.4.1
Cancel the common factor.
Step 17.1.4.2.4.2
Rewrite the expression.
Step 17.1.4.2.5
Evaluate the exponent.
Step 17.1.5
Raise to the power of .
Step 17.1.6
Multiply by .
Step 17.1.7
Cancel the common factor of .
Step 17.1.7.1
Factor out of .
Step 17.1.7.2
Cancel the common factor.
Step 17.1.7.3
Rewrite the expression.
Step 17.1.8
Multiply by .
Step 17.2
Add and .
Step 18
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 19
Step 19.1
Replace the variable with in the expression.
Step 19.2
Simplify the result.
Step 19.2.1
Simplify each term.
Step 19.2.1.1
Use the power rule to distribute the exponent.
Step 19.2.1.1.1
Apply the product rule to .
Step 19.2.1.1.2
Apply the product rule to .
Step 19.2.1.1.3
Apply the product rule to .
Step 19.2.1.2
Multiply by by adding the exponents.
Step 19.2.1.2.1
Move .
Step 19.2.1.2.2
Multiply by .
Step 19.2.1.2.2.1
Raise to the power of .
Step 19.2.1.2.2.2
Use the power rule to combine exponents.
Step 19.2.1.2.3
Add and .
Step 19.2.1.3
Raise to the power of .
Step 19.2.1.4
Simplify the numerator.
Step 19.2.1.4.1
Raise to the power of .
Step 19.2.1.4.2
Rewrite as .
Step 19.2.1.4.2.1
Use to rewrite as .
Step 19.2.1.4.2.2
Apply the power rule and multiply exponents, .
Step 19.2.1.4.2.3
Combine and .
Step 19.2.1.4.2.4
Cancel the common factor of and .
Step 19.2.1.4.2.4.1
Factor out of .
Step 19.2.1.4.2.4.2
Cancel the common factors.
Step 19.2.1.4.2.4.2.1
Factor out of .
Step 19.2.1.4.2.4.2.2
Cancel the common factor.
Step 19.2.1.4.2.4.2.3
Rewrite the expression.
Step 19.2.1.4.2.4.2.4
Divide by .
Step 19.2.1.4.3
Raise to the power of .
Step 19.2.1.5
Raise to the power of .
Step 19.2.1.6
Multiply by .
Step 19.2.1.7
Cancel the common factor of and .
Step 19.2.1.7.1
Factor out of .
Step 19.2.1.7.2
Cancel the common factors.
Step 19.2.1.7.2.1
Factor out of .
Step 19.2.1.7.2.2
Cancel the common factor.
Step 19.2.1.7.2.3
Rewrite the expression.
Step 19.2.1.8
Use the power rule to distribute the exponent.
Step 19.2.1.8.1
Apply the product rule to .
Step 19.2.1.8.2
Apply the product rule to .
Step 19.2.1.8.3
Apply the product rule to .
Step 19.2.1.9
Raise to the power of .
Step 19.2.1.10
Multiply by .
Step 19.2.1.11
Simplify the numerator.
Step 19.2.1.11.1
Raise to the power of .
Step 19.2.1.11.2
Rewrite as .
Step 19.2.1.11.2.1
Use to rewrite as .
Step 19.2.1.11.2.2
Apply the power rule and multiply exponents, .
Step 19.2.1.11.2.3
Combine and .
Step 19.2.1.11.2.4
Cancel the common factor of .
Step 19.2.1.11.2.4.1
Cancel the common factor.
Step 19.2.1.11.2.4.2
Rewrite the expression.
Step 19.2.1.11.2.5
Evaluate the exponent.
Step 19.2.1.12
Raise to the power of .
Step 19.2.1.13
Multiply by .
Step 19.2.1.14
Cancel the common factor of and .
Step 19.2.1.14.1
Factor out of .
Step 19.2.1.14.2
Cancel the common factors.
Step 19.2.1.14.2.1
Factor out of .
Step 19.2.1.14.2.2
Cancel the common factor.
Step 19.2.1.14.2.3
Rewrite the expression.
Step 19.2.1.15
Multiply .
Step 19.2.1.15.1
Combine and .
Step 19.2.1.15.2
Multiply by .
Step 19.2.2
Find the common denominator.
Step 19.2.2.1
Multiply by .
Step 19.2.2.2
Multiply by .
Step 19.2.2.3
Write as a fraction with denominator .
Step 19.2.2.4
Multiply by .
Step 19.2.2.5
Multiply by .
Step 19.2.2.6
Multiply by .
Step 19.2.3
Combine the numerators over the common denominator.
Step 19.2.4
Simplify each term.
Step 19.2.4.1
Multiply by .
Step 19.2.4.2
Multiply by .
Step 19.2.5
Simplify by adding and subtracting.
Step 19.2.5.1
Add and .
Step 19.2.5.2
Subtract from .
Step 19.2.6
The final answer is .
Step 20
These are the local extrema for .
is a local minima
is a local maxima
is a local maxima
Step 21