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Calculus Examples
Step 1
Step 1.1
Find the first derivative.
Step 1.1.1
Differentiate using the Product Rule which states that is where and .
Step 1.1.2
Differentiate using the chain rule, which states that is where and .
Step 1.1.2.1
To apply the Chain Rule, set as .
Step 1.1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.2.3
Replace all occurrences of with .
Step 1.1.3
Differentiate.
Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Simplify the expression.
Step 1.1.3.3.1
Multiply by .
Step 1.1.3.3.2
Move to the left of .
Step 1.1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.1.3.5
Multiply by .
Step 1.1.4
Simplify.
Step 1.1.4.1
Reorder terms.
Step 1.1.4.2
Reorder factors in .
Step 1.2
The first derivative of with respect to is .
Step 2
Step 2.1
Set the first derivative equal to .
Step 2.2
Factor out of .
Step 2.2.1
Factor out of .
Step 2.2.2
Multiply by .
Step 2.2.3
Factor out of .
Step 2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2.4
Set equal to and solve for .
Step 2.4.1
Set equal to .
Step 2.4.2
Solve for .
Step 2.4.2.1
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 2.4.2.2
The equation cannot be solved because is undefined.
Undefined
Step 2.4.2.3
There is no solution for
No solution
No solution
No solution
Step 2.5
Set equal to and solve for .
Step 2.5.1
Set equal to .
Step 2.5.2
Solve for .
Step 2.5.2.1
Subtract from both sides of the equation.
Step 2.5.2.2
Divide each term in by and simplify.
Step 2.5.2.2.1
Divide each term in by .
Step 2.5.2.2.2
Simplify the left side.
Step 2.5.2.2.2.1
Cancel the common factor of .
Step 2.5.2.2.2.1.1
Cancel the common factor.
Step 2.5.2.2.2.1.2
Divide by .
Step 2.5.2.2.3
Simplify the right side.
Step 2.5.2.2.3.1
Dividing two negative values results in a positive value.
Step 2.6
The final solution is all the values that make true.
Step 3
Step 3.1
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 4
Step 4.1
Evaluate at .
Step 4.1.1
Substitute for .
Step 4.1.2
Simplify.
Step 4.1.2.1
Cancel the common factor of .
Step 4.1.2.1.1
Factor out of .
Step 4.1.2.1.2
Cancel the common factor.
Step 4.1.2.1.3
Rewrite the expression.
Step 4.1.2.2
Rewrite the expression using the negative exponent rule .
Step 4.1.2.3
Multiply by .
Step 4.2
List all of the points.
Step 5