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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
Differentiate using the Quotient Rule which states that is where and .
Step 1.1.1.2
Differentiate.
Step 1.1.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.4
Simplify the expression.
Step 1.1.1.2.4.1
Add and .
Step 1.1.1.2.4.2
Multiply by .
Step 1.1.1.2.5
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.2.6
Differentiate using the Power Rule which states that is where .
Step 1.1.1.2.7
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.2.8
Simplify the expression.
Step 1.1.1.2.8.1
Add and .
Step 1.1.1.2.8.2
Multiply by .
Step 1.1.1.3
Simplify.
Step 1.1.1.3.1
Apply the distributive property.
Step 1.1.1.3.2
Simplify the numerator.
Step 1.1.1.3.2.1
Combine the opposite terms in .
Step 1.1.1.3.2.1.1
Subtract from .
Step 1.1.1.3.2.1.2
Subtract from .
Step 1.1.1.3.2.2
Multiply by .
Step 1.1.1.3.2.3
Subtract from .
Step 1.1.1.3.3
Move the negative in front of the fraction.
Step 1.1.2
Find the second derivative.
Step 1.1.2.1
Differentiate using the Constant Multiple Rule.
Step 1.1.2.1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.1.2
Apply basic rules of exponents.
Step 1.1.2.1.2.1
Rewrite as .
Step 1.1.2.1.2.2
Multiply the exponents in .
Step 1.1.2.1.2.2.1
Apply the power rule and multiply exponents, .
Step 1.1.2.1.2.2.2
Multiply by .
Step 1.1.2.2
Differentiate using the chain rule, which states that is where and .
Step 1.1.2.2.1
To apply the Chain Rule, set as .
Step 1.1.2.2.2
Differentiate using the Power Rule which states that is where .
Step 1.1.2.2.3
Replace all occurrences of with .
Step 1.1.2.3
Differentiate.
Step 1.1.2.3.1
Multiply by .
Step 1.1.2.3.2
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.3.3
Differentiate using the Power Rule which states that is where .
Step 1.1.2.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.3.5
Simplify the expression.
Step 1.1.2.3.5.1
Add and .
Step 1.1.2.3.5.2
Multiply by .
Step 1.1.2.4
Simplify.
Step 1.1.2.4.1
Rewrite the expression using the negative exponent rule .
Step 1.1.2.4.2
Combine and .
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
Since , there are no solutions.
No solution
No solution
No solution
Step 2
Step 2.1
Set the denominator in equal to to find where the expression is undefined.
Step 2.2
Add to both sides of the equation.
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
Simplify the denominator.
Step 4.2.1.1
Subtract from .
Step 4.2.1.2
Raise to the power of .
Step 4.2.2
Divide by .
Step 4.2.3
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
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Simplify the denominator.
Step 5.2.1.1
Subtract from .
Step 5.2.1.2
Raise to the power of .
Step 5.2.2
Divide by .
Step 5.2.3
The final answer is .
Step 5.3
The graph is concave up on the interval because is positive.
Concave up on since is positive
Concave up on since is positive
Step 6
The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.
Concave down on since is negative
Concave up on since is positive
Step 7