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Algebra Examples
Step 1
Step 1.1
Use to rewrite as .
Step 1.2
Differentiate using the Product Rule which states that is where and .
Step 1.3
Differentiate using the chain rule, which states that is where and .
Step 1.3.1
To apply the Chain Rule, set as .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Replace all occurrences of with .
Step 1.4
To write as a fraction with a common denominator, multiply by .
Step 1.5
Combine and .
Step 1.6
Combine the numerators over the common denominator.
Step 1.7
Simplify the numerator.
Step 1.7.1
Multiply by .
Step 1.7.2
Subtract from .
Step 1.8
Combine fractions.
Step 1.8.1
Move the negative in front of the fraction.
Step 1.8.2
Combine and .
Step 1.8.3
Move to the denominator using the negative exponent rule .
Step 1.8.4
Combine and .
Step 1.9
By the Sum Rule, the derivative of with respect to is .
Step 1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.11
Add and .
Step 1.12
Since is constant with respect to , the derivative of with respect to is .
Step 1.13
Differentiate using the Power Rule which states that is where .
Step 1.14
Combine fractions.
Step 1.14.1
Multiply by .
Step 1.14.2
Combine and .
Step 1.14.3
Simplify the expression.
Step 1.14.3.1
Move to the left of .
Step 1.14.3.2
Rewrite as .
Step 1.14.3.3
Move the negative in front of the fraction.
Step 1.15
Differentiate using the Power Rule which states that is where .
Step 1.16
Multiply by .
Step 1.17
To write as a fraction with a common denominator, multiply by .
Step 1.18
Combine and .
Step 1.19
Combine the numerators over the common denominator.
Step 1.20
Multiply by by adding the exponents.
Step 1.20.1
Move .
Step 1.20.2
Use the power rule to combine exponents.
Step 1.20.3
Combine the numerators over the common denominator.
Step 1.20.4
Add and .
Step 1.20.5
Divide by .
Step 1.21
Simplify .
Step 1.22
Move to the left of .
Step 1.23
Simplify.
Step 1.23.1
Apply the distributive property.
Step 1.23.2
Simplify the numerator.
Step 1.23.2.1
Simplify each term.
Step 1.23.2.1.1
Multiply by .
Step 1.23.2.1.2
Multiply by .
Step 1.23.2.2
Subtract from .
Step 1.23.3
Factor out of .
Step 1.23.4
Rewrite as .
Step 1.23.5
Factor out of .
Step 1.23.6
Rewrite as .
Step 1.23.7
Move the negative in front of the fraction.
Step 2
Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Multiply the exponents in .
Step 2.3.1
Apply the power rule and multiply exponents, .
Step 2.3.2
Cancel the common factor of .
Step 2.3.2.1
Cancel the common factor.
Step 2.3.2.2
Rewrite the expression.
Step 2.4
Simplify.
Step 2.5
Differentiate.
Step 2.5.1
By the Sum Rule, the derivative of with respect to is .
Step 2.5.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.3
Differentiate using the Power Rule which states that is where .
Step 2.5.4
Multiply by .
Step 2.5.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.6
Simplify the expression.
Step 2.5.6.1
Add and .
Step 2.5.6.2
Move to the left of .
Step 2.6
Differentiate using the chain rule, which states that is where and .
Step 2.6.1
To apply the Chain Rule, set as .
Step 2.6.2
Differentiate using the Power Rule which states that is where .
Step 2.6.3
Replace all occurrences of with .
Step 2.7
To write as a fraction with a common denominator, multiply by .
Step 2.8
Combine and .
Step 2.9
Combine the numerators over the common denominator.
Step 2.10
Simplify the numerator.
Step 2.10.1
Multiply by .
Step 2.10.2
Subtract from .
Step 2.11
Combine fractions.
Step 2.11.1
Move the negative in front of the fraction.
Step 2.11.2
Combine and .
Step 2.11.3
Move to the denominator using the negative exponent rule .
Step 2.12
By the Sum Rule, the derivative of with respect to is .
Step 2.13
Since is constant with respect to , the derivative of with respect to is .
Step 2.14
Add and .
Step 2.15
Since is constant with respect to , the derivative of with respect to is .
Step 2.16
Multiply.
Step 2.16.1
Multiply by .
Step 2.16.2
Multiply by .
Step 2.17
Differentiate using the Power Rule which states that is where .
Step 2.18
Combine fractions.
Step 2.18.1
Multiply by .
Step 2.18.2
Multiply by .
Step 2.18.3
Move to the left of .
Step 2.19
Simplify.
Step 2.19.1
Apply the distributive property.
Step 2.19.2
Simplify the numerator.
Step 2.19.2.1
Let . Substitute for all occurrences of .
Step 2.19.2.1.1
Rewrite using the commutative property of multiplication.
Step 2.19.2.1.2
Multiply by by adding the exponents.
Step 2.19.2.1.2.1
Move .
Step 2.19.2.1.2.2
Multiply by .
Step 2.19.2.1.3
Multiply by .
Step 2.19.2.2
Replace all occurrences of with .
Step 2.19.2.3
Simplify.
Step 2.19.2.3.1
Simplify each term.
Step 2.19.2.3.1.1
Multiply the exponents in .
Step 2.19.2.3.1.1.1
Apply the power rule and multiply exponents, .
Step 2.19.2.3.1.1.2
Cancel the common factor of .
Step 2.19.2.3.1.1.2.1
Cancel the common factor.
Step 2.19.2.3.1.1.2.2
Rewrite the expression.
Step 2.19.2.3.1.2
Simplify.
Step 2.19.2.3.1.3
Apply the distributive property.
Step 2.19.2.3.1.4
Multiply by .
Step 2.19.2.3.1.5
Multiply by .
Step 2.19.2.3.2
Subtract from .
Step 2.19.2.3.3
Add and .
Step 2.19.3
Combine terms.
Step 2.19.3.1
Multiply by .
Step 2.19.3.2
Multiply by .
Step 2.19.3.3
Rewrite as a product.
Step 2.19.3.4
Multiply by .
Step 2.19.4
Simplify the denominator.
Step 2.19.4.1
Factor out of .
Step 2.19.4.1.1
Factor out of .
Step 2.19.4.1.2
Factor out of .
Step 2.19.4.1.3
Factor out of .
Step 2.19.4.2
Combine exponents.
Step 2.19.4.2.1
Multiply by .
Step 2.19.4.2.2
Raise to the power of .
Step 2.19.4.2.3
Use the power rule to combine exponents.
Step 2.19.4.2.4
Write as a fraction with a common denominator.
Step 2.19.4.2.5
Combine the numerators over the common denominator.
Step 2.19.4.2.6
Add and .
Step 2.19.5
Factor out of .
Step 2.19.6
Rewrite as .
Step 2.19.7
Factor out of .
Step 2.19.8
Rewrite as .
Step 2.19.9
Move the negative in front of the fraction.
Step 2.19.10
Multiply by .
Step 2.19.11
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
Use to rewrite as .
Step 4.1.2
Differentiate using the Product Rule which states that is where and .
Step 4.1.3
Differentiate using the chain rule, which states that is where and .
Step 4.1.3.1
To apply the Chain Rule, set as .
Step 4.1.3.2
Differentiate using the Power Rule which states that is where .
Step 4.1.3.3
Replace all occurrences of with .
Step 4.1.4
To write as a fraction with a common denominator, multiply by .
Step 4.1.5
Combine and .
Step 4.1.6
Combine the numerators over the common denominator.
Step 4.1.7
Simplify the numerator.
Step 4.1.7.1
Multiply by .
Step 4.1.7.2
Subtract from .
Step 4.1.8
Combine fractions.
Step 4.1.8.1
Move the negative in front of the fraction.
Step 4.1.8.2
Combine and .
Step 4.1.8.3
Move to the denominator using the negative exponent rule .
Step 4.1.8.4
Combine and .
Step 4.1.9
By the Sum Rule, the derivative of with respect to is .
Step 4.1.10
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.11
Add and .
Step 4.1.12
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.13
Differentiate using the Power Rule which states that is where .
Step 4.1.14
Combine fractions.
Step 4.1.14.1
Multiply by .
Step 4.1.14.2
Combine and .
Step 4.1.14.3
Simplify the expression.
Step 4.1.14.3.1
Move to the left of .
Step 4.1.14.3.2
Rewrite as .
Step 4.1.14.3.3
Move the negative in front of the fraction.
Step 4.1.15
Differentiate using the Power Rule which states that is where .
Step 4.1.16
Multiply by .
Step 4.1.17
To write as a fraction with a common denominator, multiply by .
Step 4.1.18
Combine and .
Step 4.1.19
Combine the numerators over the common denominator.
Step 4.1.20
Multiply by by adding the exponents.
Step 4.1.20.1
Move .
Step 4.1.20.2
Use the power rule to combine exponents.
Step 4.1.20.3
Combine the numerators over the common denominator.
Step 4.1.20.4
Add and .
Step 4.1.20.5
Divide by .
Step 4.1.21
Simplify .
Step 4.1.22
Move to the left of .
Step 4.1.23
Simplify.
Step 4.1.23.1
Apply the distributive property.
Step 4.1.23.2
Simplify the numerator.
Step 4.1.23.2.1
Simplify each term.
Step 4.1.23.2.1.1
Multiply by .
Step 4.1.23.2.1.2
Multiply by .
Step 4.1.23.2.2
Subtract from .
Step 4.1.23.3
Factor out of .
Step 4.1.23.4
Rewrite as .
Step 4.1.23.5
Factor out of .
Step 4.1.23.6
Rewrite as .
Step 4.1.23.7
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
Set the numerator equal to zero.
Step 5.3
Solve the equation for .
Step 5.3.1
Add to 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
Cancel the common factor of .
Step 5.3.2.2.1.1
Cancel the common factor.
Step 5.3.2.2.1.2
Divide by .
Step 6
Step 6.1
Convert expressions with fractional exponents to radicals.
Step 6.1.1
Apply the rule to rewrite the exponentiation as a radical.
Step 6.1.2
Anything raised to is the base itself.
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, square 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
Simplify .
Step 6.3.2.2.1.1
Apply the product rule to .
Step 6.3.2.2.1.2
Raise to the power of .
Step 6.3.2.2.1.3
Multiply the exponents in .
Step 6.3.2.2.1.3.1
Apply the power rule and multiply exponents, .
Step 6.3.2.2.1.3.2
Cancel the common factor of .
Step 6.3.2.2.1.3.2.1
Cancel the common factor.
Step 6.3.2.2.1.3.2.2
Rewrite the expression.
Step 6.3.2.2.1.4
Simplify.
Step 6.3.2.2.1.5
Apply the distributive property.
Step 6.3.2.2.1.6
Multiply.
Step 6.3.2.2.1.6.1
Multiply by .
Step 6.3.2.2.1.6.2
Multiply by .
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
Subtract from both sides of the equation.
Step 6.3.3.2
Divide each term in by and simplify.
Step 6.3.3.2.1
Divide each term in by .
Step 6.3.3.2.2
Simplify the left side.
Step 6.3.3.2.2.1
Cancel the common factor of .
Step 6.3.3.2.2.1.1
Cancel the common factor.
Step 6.3.3.2.2.1.2
Divide by .
Step 6.3.3.2.3
Simplify the right side.
Step 6.3.3.2.3.1
Divide by .
Step 6.4
Set the radicand in less than to find where the expression is undefined.
Step 6.5
Solve for .
Step 6.5.1
Subtract from both sides of the inequality.
Step 6.5.2
Divide each term in by and simplify.
Step 6.5.2.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 6.5.2.2
Simplify the left side.
Step 6.5.2.2.1
Dividing two negative values results in a positive value.
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 6.6
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
Simplify the numerator.
Step 9.1.1
Cancel the common factor of .
Step 9.1.1.1
Cancel the common factor.
Step 9.1.1.2
Rewrite the expression.
Step 9.1.2
Subtract from .
Step 9.2
Simplify the denominator.
Step 9.2.1
To write as a fraction with a common denominator, multiply by .
Step 9.2.2
Combine and .
Step 9.2.3
Combine the numerators over the common denominator.
Step 9.2.4
Simplify the numerator.
Step 9.2.4.1
Multiply by .
Step 9.2.4.2
Subtract from .
Step 9.2.5
Apply the product rule to .
Step 9.3
Combine and .
Step 9.4
Simplify the numerator.
Step 9.4.1
Rewrite as .
Step 9.4.2
Use the power rule to combine exponents.
Step 9.4.3
To write as a fraction with a common denominator, multiply by .
Step 9.4.4
Combine and .
Step 9.4.5
Combine the numerators over the common denominator.
Step 9.4.6
Simplify the numerator.
Step 9.4.6.1
Multiply by .
Step 9.4.6.2
Add and .
Step 9.5
Multiply the numerator by the reciprocal of the denominator.
Step 9.6
Combine and .
Step 9.7
Move the negative in front of the fraction.
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
To write as a fraction with a common denominator, multiply by .
Step 11.2.2
Combine and .
Step 11.2.3
Combine the numerators over the common denominator.
Step 11.2.4
Simplify the numerator.
Step 11.2.4.1
Multiply by .
Step 11.2.4.2
Subtract from .
Step 11.2.5
Rewrite as .
Step 11.2.6
Multiply by .
Step 11.2.7
Combine and simplify the denominator.
Step 11.2.7.1
Multiply by .
Step 11.2.7.2
Raise to the power of .
Step 11.2.7.3
Raise to the power of .
Step 11.2.7.4
Use the power rule to combine exponents.
Step 11.2.7.5
Add and .
Step 11.2.7.6
Rewrite as .
Step 11.2.7.6.1
Use to rewrite as .
Step 11.2.7.6.2
Apply the power rule and multiply exponents, .
Step 11.2.7.6.3
Combine and .
Step 11.2.7.6.4
Cancel the common factor of .
Step 11.2.7.6.4.1
Cancel the common factor.
Step 11.2.7.6.4.2
Rewrite the expression.
Step 11.2.7.6.5
Evaluate the exponent.
Step 11.2.8
Simplify the numerator.
Step 11.2.8.1
Combine using the product rule for radicals.
Step 11.2.8.2
Multiply by .
Step 11.2.9
Multiply .
Step 11.2.9.1
Multiply by .
Step 11.2.9.2
Multiply by .
Step 11.2.10
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 expression.
Step 13.1.1
Multiply by .
Step 13.1.2
Subtract from .
Step 13.1.3
Rewrite as .
Step 13.1.4
Apply the power rule and multiply exponents, .
Step 13.2
Cancel the common factor of .
Step 13.2.1
Cancel the common factor.
Step 13.2.2
Rewrite the expression.
Step 13.3
Simplify the expression.
Step 13.3.1
Raising to any positive power yields .
Step 13.3.2
Multiply by .
Step 13.3.3
The expression contains a division by . The expression is undefined.
Undefined
Step 13.4
The expression contains a division by . The expression is undefined.
Undefined
Undefined
Step 14
Since the first derivative test failed, there are no local extrema.
No Local Extrema
Step 15