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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
Differentiate using the Constant Rule.
Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Add and .
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
Subtract from both sides of the equation.
Step 5.3
Divide each term in by and simplify.
Step 5.3.1
Divide each term in by .
Step 5.3.2
Simplify the left side.
Step 5.3.2.1
Cancel the common factor of .
Step 5.3.2.1.1
Cancel the common factor.
Step 5.3.2.1.2
Divide by .
Step 5.3.3
Simplify the right side.
Step 5.3.3.1
Cancel the common factor of and .
Step 5.3.3.1.1
Factor out of .
Step 5.3.3.1.2
Cancel the common factors.
Step 5.3.3.1.2.1
Factor out of .
Step 5.3.3.1.2.2
Cancel the common factor.
Step 5.3.3.1.2.3
Rewrite the expression.
Step 5.3.3.2
Move the negative in front of the fraction.
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
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 10
Step 10.1
Replace the variable with in the expression.
Step 10.2
Simplify the result.
Step 10.2.1
Simplify each term.
Step 10.2.1.1
Use the power rule to distribute the exponent.
Step 10.2.1.1.1
Apply the product rule to .
Step 10.2.1.1.2
Apply the product rule to .
Step 10.2.1.2
Raise to the power of .
Step 10.2.1.3
Multiply by .
Step 10.2.1.4
Raise to the power of .
Step 10.2.1.5
Raise to the power of .
Step 10.2.1.6
Cancel the common factor of .
Step 10.2.1.6.1
Factor out of .
Step 10.2.1.6.2
Cancel the common factor.
Step 10.2.1.6.3
Rewrite the expression.
Step 10.2.1.7
Multiply .
Step 10.2.1.7.1
Multiply by .
Step 10.2.1.7.2
Combine and .
Step 10.2.1.7.3
Multiply by .
Step 10.2.1.8
Move the negative in front of the fraction.
Step 10.2.2
Combine fractions.
Step 10.2.2.1
Combine the numerators over the common denominator.
Step 10.2.2.2
Simplify the expression.
Step 10.2.2.2.1
Subtract from .
Step 10.2.2.2.2
Move the negative in front of the fraction.
Step 10.2.3
To write as a fraction with a common denominator, multiply by .
Step 10.2.4
Combine and .
Step 10.2.5
Combine the numerators over the common denominator.
Step 10.2.6
Simplify the numerator.
Step 10.2.6.1
Multiply by .
Step 10.2.6.2
Subtract from .
Step 10.2.7
Move the negative in front of the fraction.
Step 10.2.8
The final answer is .
Step 11
These are the local extrema for .
is a local minima
Step 12