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Calculus Examples
Step 1
Write as a function.
Step 2
Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
Step 2.2.1
Differentiate using the chain rule, which states that is where and .
Step 2.2.1.1
To apply the Chain Rule, set as .
Step 2.2.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.2.1.3
Replace all occurrences of with .
Step 2.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.3
Differentiate using the Power Rule which states that is where .
Step 2.2.4
Multiply by .
Step 2.2.5
Move to the left of .
Step 2.3
Evaluate .
Step 2.3.1
Differentiate using the chain rule, which states that is where and .
Step 2.3.1.1
To apply the Chain Rule, set as .
Step 2.3.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3.1.3
Replace all occurrences of with .
Step 2.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.3
Differentiate using the Power Rule which states that is where .
Step 2.3.4
Multiply by .
Step 2.3.5
Move to the left of .
Step 2.3.6
Rewrite as .
Step 3
Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
Step 3.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.2
Differentiate using the chain rule, which states that is where and .
Step 3.2.2.1
To apply the Chain Rule, set as .
Step 3.2.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.2.2.3
Replace all occurrences of with .
Step 3.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.4
Differentiate using the Power Rule which states that is where .
Step 3.2.5
Multiply by .
Step 3.2.6
Move to the left of .
Step 3.2.7
Multiply by .
Step 3.3
Evaluate .
Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Differentiate using the chain rule, which states that is where and .
Step 3.3.2.1
To apply the Chain Rule, set as .
Step 3.3.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.3.2.3
Replace all occurrences of with .
Step 3.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.4
Differentiate using the Power Rule which states that is where .
Step 3.3.5
Multiply by .
Step 3.3.6
Move to the left of .
Step 3.3.7
Rewrite as .
Step 3.3.8
Multiply by .
Step 3.3.9
Multiply by .
Step 4
To find the local maximum and minimum values of the function, set the derivative equal to and solve.
Step 5
Step 5.1
Find the first derivative.
Step 5.1.1
By the Sum Rule, the derivative of with respect to is .
Step 5.1.2
Evaluate .
Step 5.1.2.1
Differentiate using the chain rule, which states that is where and .
Step 5.1.2.1.1
To apply the Chain Rule, set as .
Step 5.1.2.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 5.1.2.1.3
Replace all occurrences of with .
Step 5.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.2.3
Differentiate using the Power Rule which states that is where .
Step 5.1.2.4
Multiply by .
Step 5.1.2.5
Move to the left of .
Step 5.1.3
Evaluate .
Step 5.1.3.1
Differentiate using the chain rule, which states that is where and .
Step 5.1.3.1.1
To apply the Chain Rule, set as .
Step 5.1.3.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 5.1.3.1.3
Replace all occurrences of with .
Step 5.1.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3.3
Differentiate using the Power Rule which states that is where .
Step 5.1.3.4
Multiply by .
Step 5.1.3.5
Move to the left of .
Step 5.1.3.6
Rewrite as .
Step 5.2
The first derivative of with respect to is .
Step 6
Step 6.1
Set the first derivative equal to .
Step 6.2
Move to the right side of the equation by adding it to both sides.
Step 6.3
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 6.4
Expand the left side.
Step 6.4.1
Rewrite as .
Step 6.4.2
Expand by moving outside the logarithm.
Step 6.4.3
The natural logarithm of is .
Step 6.4.4
Multiply by .
Step 6.5
Expand the right side.
Step 6.5.1
Expand by moving outside the logarithm.
Step 6.5.2
The natural logarithm of is .
Step 6.5.3
Multiply by .
Step 6.6
Move all terms containing to the left side of the equation.
Step 6.6.1
Add to both sides of the equation.
Step 6.6.2
Add and .
Step 6.7
Subtract from both sides of the equation.
Step 6.8
Divide each term in by and simplify.
Step 6.8.1
Divide each term in by .
Step 6.8.2
Simplify the left side.
Step 6.8.2.1
Cancel the common factor of .
Step 6.8.2.1.1
Cancel the common factor.
Step 6.8.2.1.2
Divide by .
Step 6.8.3
Simplify the right side.
Step 6.8.3.1
Move the negative in front of the fraction.
Step 7
Step 7.1
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 8
Critical points to evaluate.
Step 9
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 10
Step 10.1
Rewrite as .
Step 10.2
Simplify by moving inside the logarithm.
Step 10.3
Multiply .
Step 10.3.1
Multiply by .
Step 10.3.2
Simplify by moving inside the logarithm.
Step 10.4
Simplify by moving inside the logarithm.
Step 10.5
Exponentiation and log are inverse functions.
Step 10.6
Multiply the exponents in .
Step 10.6.1
Apply the power rule and multiply exponents, .
Step 10.6.2
Multiply by .
Step 10.7
Multiply the exponents in .
Step 10.7.1
Apply the power rule and multiply exponents, .
Step 10.7.2
Combine and .
Step 10.7.3
Move the negative in front of the fraction.
Step 10.8
Rewrite the expression using the negative exponent rule .
Step 10.9
Combine and .
Step 10.10
Rewrite as .
Step 10.11
Simplify by moving inside the logarithm.
Step 10.12
Multiply .
Step 10.12.1
Multiply by .
Step 10.12.2
Multiply by .
Step 10.13
Exponentiation and log are inverse functions.
Step 11
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 12
Step 12.1
Simplify to substitute in .
Step 12.1.1
Rewrite as .
Step 12.1.2
Simplify by moving inside the logarithm.
Step 12.2
Replace the variable with in the expression.
Step 12.3
Simplify the result.
Step 12.3.1
Simplify each term.
Step 12.3.1.1
Multiply .
Step 12.3.1.1.1
Multiply by .
Step 12.3.1.1.2
Simplify by moving inside the logarithm.
Step 12.3.1.2
Simplify by moving inside the logarithm.
Step 12.3.1.3
Exponentiation and log are inverse functions.
Step 12.3.1.4
Multiply the exponents in .
Step 12.3.1.4.1
Apply the power rule and multiply exponents, .
Step 12.3.1.4.2
Multiply by .
Step 12.3.1.5
Multiply the exponents in .
Step 12.3.1.5.1
Apply the power rule and multiply exponents, .
Step 12.3.1.5.2
Combine and .
Step 12.3.1.5.3
Move the negative in front of the fraction.
Step 12.3.1.6
Rewrite the expression using the negative exponent rule .
Step 12.3.1.7
Multiply .
Step 12.3.1.7.1
Multiply by .
Step 12.3.1.7.2
Multiply by .
Step 12.3.1.8
Exponentiation and log are inverse functions.
Step 12.3.2
The final answer is .
Step 13
These are the local extrema for .
is a local minima
Step 14