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Calculus Examples
Step 1
Step 1.1
Find the second derivative.
Step 1.1.1
Find the first derivative.
Step 1.1.1.1
Use to rewrite as .
Step 1.1.1.2
Differentiate using the Product Rule which states that is where and .
Step 1.1.1.3
Differentiate using the chain rule, which states that is where and .
Step 1.1.1.3.1
To apply the Chain Rule, set as .
Step 1.1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.3.3
Replace all occurrences of with .
Step 1.1.1.4
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.5
Combine and .
Step 1.1.1.6
Combine the numerators over the common denominator.
Step 1.1.1.7
Simplify the numerator.
Step 1.1.1.7.1
Multiply by .
Step 1.1.1.7.2
Subtract from .
Step 1.1.1.8
Combine fractions.
Step 1.1.1.8.1
Move the negative in front of the fraction.
Step 1.1.1.8.2
Combine and .
Step 1.1.1.8.3
Move to the denominator using the negative exponent rule .
Step 1.1.1.8.4
Combine and .
Step 1.1.1.9
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.11
Add and .
Step 1.1.1.12
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.13
Differentiate using the Power Rule which states that is where .
Step 1.1.1.14
Combine fractions.
Step 1.1.1.14.1
Multiply by .
Step 1.1.1.14.2
Combine and .
Step 1.1.1.14.3
Simplify the expression.
Step 1.1.1.14.3.1
Move to the left of .
Step 1.1.1.14.3.2
Rewrite as .
Step 1.1.1.14.3.3
Move the negative in front of the fraction.
Step 1.1.1.15
Differentiate using the Power Rule which states that is where .
Step 1.1.1.16
Multiply by .
Step 1.1.1.17
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.18
Combine and .
Step 1.1.1.19
Combine the numerators over the common denominator.
Step 1.1.1.20
Multiply by by adding the exponents.
Step 1.1.1.20.1
Move .
Step 1.1.1.20.2
Use the power rule to combine exponents.
Step 1.1.1.20.3
Combine the numerators over the common denominator.
Step 1.1.1.20.4
Add and .
Step 1.1.1.20.5
Divide by .
Step 1.1.1.21
Simplify .
Step 1.1.1.22
Move to the left of .
Step 1.1.1.23
Simplify.
Step 1.1.1.23.1
Apply the distributive property.
Step 1.1.1.23.2
Simplify the numerator.
Step 1.1.1.23.2.1
Simplify each term.
Step 1.1.1.23.2.1.1
Multiply by .
Step 1.1.1.23.2.1.2
Multiply by .
Step 1.1.1.23.2.2
Subtract from .
Step 1.1.1.23.3
Factor out of .
Step 1.1.1.23.3.1
Factor out of .
Step 1.1.1.23.3.2
Factor out of .
Step 1.1.1.23.3.3
Factor out of .
Step 1.1.1.23.4
Factor out of .
Step 1.1.1.23.5
Rewrite as .
Step 1.1.1.23.6
Factor out of .
Step 1.1.1.23.7
Rewrite as .
Step 1.1.1.23.8
Move the negative in front of the fraction.
Step 1.1.2
Find the second derivative.
Step 1.1.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.2
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.2.3
Multiply the exponents in .
Step 1.1.2.3.1
Apply the power rule and multiply exponents, .
Step 1.1.2.3.2
Cancel the common factor of .
Step 1.1.2.3.2.1
Cancel the common factor.
Step 1.1.2.3.2.2
Rewrite the expression.
Step 1.1.2.4
Simplify.
Step 1.1.2.5
Differentiate.
Step 1.1.2.5.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.5.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.5.4
Simplify the expression.
Step 1.1.2.5.4.1
Add and .
Step 1.1.2.5.4.2
Multiply by .
Step 1.1.2.6
Differentiate using the chain rule, which states that is where and .
Step 1.1.2.6.1
To apply the Chain Rule, set as .
Step 1.1.2.6.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.6.3
Replace all occurrences of with .
Step 1.1.2.7
To write as a fraction with a common denominator, multiply by .
Step 1.1.2.8
Combine and .
Step 1.1.2.9
Combine the numerators over the common denominator.
Step 1.1.2.10
Simplify the numerator.
Step 1.1.2.10.1
Multiply by .
Step 1.1.2.10.2
Subtract from .
Step 1.1.2.11
Combine fractions.
Step 1.1.2.11.1
Move the negative in front of the fraction.
Step 1.1.2.11.2
Combine and .
Step 1.1.2.11.3
Move to the denominator using the negative exponent rule .
Step 1.1.2.12
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.13
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.14
Add and .
Step 1.1.2.15
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.16
Multiply.
Step 1.1.2.16.1
Multiply by .
Step 1.1.2.16.2
Multiply by .
Step 1.1.2.17
Differentiate using the Power Rule which states that is where .
Step 1.1.2.18
Combine fractions.
Step 1.1.2.18.1
Multiply by .
Step 1.1.2.18.2
Multiply by .
Step 1.1.2.18.3
Reorder.
Step 1.1.2.18.3.1
Move to the left of .
Step 1.1.2.18.3.2
Move to the left of .
Step 1.1.2.19
Simplify.
Step 1.1.2.19.1
Apply the distributive property.
Step 1.1.2.19.2
Apply the distributive property.
Step 1.1.2.19.3
Simplify the numerator.
Step 1.1.2.19.3.1
Factor out of .
Step 1.1.2.19.3.1.1
Factor out of .
Step 1.1.2.19.3.1.2
Factor out of .
Step 1.1.2.19.3.1.3
Factor out of .
Step 1.1.2.19.3.2
Let . Substitute for all occurrences of .
Step 1.1.2.19.3.2.1
Rewrite using the commutative property of multiplication.
Step 1.1.2.19.3.2.2
Multiply by by adding the exponents.
Step 1.1.2.19.3.2.2.1
Move .
Step 1.1.2.19.3.2.2.2
Multiply by .
Step 1.1.2.19.3.3
Replace all occurrences of with .
Step 1.1.2.19.3.4
Simplify.
Step 1.1.2.19.3.4.1
Simplify each term.
Step 1.1.2.19.3.4.1.1
Multiply the exponents in .
Step 1.1.2.19.3.4.1.1.1
Apply the power rule and multiply exponents, .
Step 1.1.2.19.3.4.1.1.2
Cancel the common factor of .
Step 1.1.2.19.3.4.1.1.2.1
Cancel the common factor.
Step 1.1.2.19.3.4.1.1.2.2
Rewrite the expression.
Step 1.1.2.19.3.4.1.2
Simplify.
Step 1.1.2.19.3.4.1.3
Apply the distributive property.
Step 1.1.2.19.3.4.1.4
Multiply by .
Step 1.1.2.19.3.4.1.5
Multiply by .
Step 1.1.2.19.3.4.2
Subtract from .
Step 1.1.2.19.3.4.3
Add and .
Step 1.1.2.19.4
Combine terms.
Step 1.1.2.19.4.1
Combine and .
Step 1.1.2.19.4.2
Multiply by .
Step 1.1.2.19.4.3
Multiply by .
Step 1.1.2.19.4.4
Rewrite as a product.
Step 1.1.2.19.4.5
Multiply by .
Step 1.1.2.19.5
Simplify the denominator.
Step 1.1.2.19.5.1
Factor out of .
Step 1.1.2.19.5.1.1
Factor out of .
Step 1.1.2.19.5.1.2
Factor out of .
Step 1.1.2.19.5.1.3
Factor out of .
Step 1.1.2.19.5.2
Combine exponents.
Step 1.1.2.19.5.2.1
Multiply by .
Step 1.1.2.19.5.2.2
Raise to the power of .
Step 1.1.2.19.5.2.3
Use the power rule to combine exponents.
Step 1.1.2.19.5.2.4
Write as a fraction with a common denominator.
Step 1.1.2.19.5.2.5
Combine the numerators over the common denominator.
Step 1.1.2.19.5.2.6
Add and .
Step 1.1.2.19.6
Factor out of .
Step 1.1.2.19.7
Rewrite as .
Step 1.1.2.19.8
Factor out of .
Step 1.1.2.19.9
Rewrite as .
Step 1.1.2.19.10
Move the negative in front of the fraction.
Step 1.1.2.19.11
Multiply by .
Step 1.1.2.19.12
Multiply by .
Step 1.1.3
The second derivative of with respect to is .
Step 1.2
Set the second derivative equal to then solve the equation .
Step 1.2.1
Set the second derivative equal to .
Step 1.2.2
Set the numerator equal to zero.
Step 1.2.3
Solve the equation for .
Step 1.2.3.1
Divide each term in by and simplify.
Step 1.2.3.1.1
Divide each term in by .
Step 1.2.3.1.2
Simplify the left side.
Step 1.2.3.1.2.1
Cancel the common factor of .
Step 1.2.3.1.2.1.1
Cancel the common factor.
Step 1.2.3.1.2.1.2
Divide by .
Step 1.2.3.1.3
Simplify the right side.
Step 1.2.3.1.3.1
Divide by .
Step 1.2.3.2
Add to both sides of the equation.
Step 1.2.4
Exclude the solutions that do not make true.
Step 2
Step 2.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 2.2
Solve for .
Step 2.2.1
Subtract from both sides of the inequality.
Step 2.2.2
Divide each term in by and simplify.
Step 2.2.2.1
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 2.2.2.2
Simplify the left side.
Step 2.2.2.2.1
Dividing two negative values results in a positive value.
Step 2.2.2.2.2
Divide by .
Step 2.2.2.3
Simplify the right side.
Step 2.2.2.3.1
Divide by .
Step 2.3
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Interval Notation:
Set-Builder Notation:
Step 3
Create intervals around the -values where the second derivative is zero or undefined.
Step 4
Step 4.1
Replace the variable with in the expression.
Step 4.2
Simplify the result.
Step 4.2.1
Factor out of .
Step 4.2.2
Cancel the common factors.
Step 4.2.2.1
Factor out of .
Step 4.2.2.2
Cancel the common factor.
Step 4.2.2.3
Rewrite the expression.
Step 4.2.3
Subtract from .
Step 4.2.4
Subtract from .
Step 4.2.5
Multiply by .
Step 4.2.6
Move the negative in front of the fraction.
Step 4.2.7
Move to the denominator using the negative exponent rule .
Step 4.2.8
Multiply by by adding the exponents.
Step 4.2.8.1
Use the power rule to combine exponents.
Step 4.2.8.2
To write as a fraction with a common denominator, multiply by .
Step 4.2.8.3
Combine and .
Step 4.2.8.4
Combine the numerators over the common denominator.
Step 4.2.8.5
Simplify the numerator.
Step 4.2.8.5.1
Multiply by .
Step 4.2.8.5.2
Subtract from .
Step 4.2.9
The final answer is .
Step 4.3
The graph is concave down on the interval because is negative.
Concave down on since is negative
Concave down on since is negative
Step 5