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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 chain rule, which states that is where and .
Step 1.1.1.1.1
To apply the Chain Rule, set as .
Step 1.1.1.1.2
Differentiate using the Power Rule which states that is where .
Step 1.1.1.1.3
Replace all occurrences of with .
Step 1.1.1.2
To write as a fraction with a common denominator, multiply by .
Step 1.1.1.3
Combine and .
Step 1.1.1.4
Combine the numerators over the common denominator.
Step 1.1.1.5
Simplify the numerator.
Step 1.1.1.5.1
Multiply by .
Step 1.1.1.5.2
Subtract from .
Step 1.1.1.6
Combine fractions.
Step 1.1.1.6.1
Move the negative in front of the fraction.
Step 1.1.1.6.2
Combine and .
Step 1.1.1.6.3
Move to the denominator using the negative exponent rule .
Step 1.1.1.7
By the Sum Rule, the derivative of with respect to is .
Step 1.1.1.8
Differentiate using the Power Rule which states that is where .
Step 1.1.1.9
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.1.10
Simplify the expression.
Step 1.1.1.10.1
Add and .
Step 1.1.1.10.2
Multiply by .
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
Combine and .
Step 1.1.2.1.2.2.3
Move the negative in front of the fraction.
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
To write as a fraction with a common denominator, multiply by .
Step 1.1.2.4
Combine and .
Step 1.1.2.5
Combine the numerators over the common denominator.
Step 1.1.2.6
Simplify the numerator.
Step 1.1.2.6.1
Multiply by .
Step 1.1.2.6.2
Subtract from .
Step 1.1.2.7
Combine fractions.
Step 1.1.2.7.1
Move the negative in front of the fraction.
Step 1.1.2.7.2
Combine and .
Step 1.1.2.7.3
Move to the denominator using the negative exponent rule .
Step 1.1.2.7.4
Multiply by .
Step 1.1.2.7.5
Multiply by .
Step 1.1.2.8
By the Sum Rule, the derivative of with respect to is .
Step 1.1.2.9
Differentiate using the Power Rule which states that is where .
Step 1.1.2.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2.11
Simplify the expression.
Step 1.1.2.11.1
Add and .
Step 1.1.2.11.2
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
Since , there are no solutions.
No solution
No solution
No solution
Step 2
Step 2.1
Apply the rule to rewrite the exponentiation as a radical.
Step 2.2
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.
Interval Notation:
Set-Builder Notation:
Interval Notation:
Set-Builder Notation:
Step 3
The graph is concave down because the second derivative is negative.
The graph is concave down
Step 4