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Calculus Examples
Step 1
Differentiate both sides of the equation.
Step 2
Step 2.1
Differentiate.
Step 2.1.1
By the Sum Rule, the derivative of with respect to is .
Step 2.1.2
Differentiate using the Power Rule which states that is where .
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 Power Rule which states that is where .
Step 2.2.1.3
Replace all occurrences of with .
Step 2.2.2
Rewrite as .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
Simplify.
Step 2.4.1
Add and .
Step 2.4.2
Reorder terms.
Step 3
Since is constant with respect to , the derivative of with respect to is .
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Step 5.1
Subtract from both sides of the equation.
Step 5.2
Divide each term in by and simplify.
Step 5.2.1
Divide each term in by .
Step 5.2.2
Simplify the left side.
Step 5.2.2.1
Cancel the common factor of .
Step 5.2.2.1.1
Cancel the common factor.
Step 5.2.2.1.2
Rewrite the expression.
Step 5.2.2.2
Cancel the common factor of .
Step 5.2.2.2.1
Cancel the common factor.
Step 5.2.2.2.2
Divide by .
Step 5.2.3
Simplify the right side.
Step 5.2.3.1
Cancel the common factor of and .
Step 5.2.3.1.1
Factor out of .
Step 5.2.3.1.2
Cancel the common factors.
Step 5.2.3.1.2.1
Factor out of .
Step 5.2.3.1.2.2
Cancel the common factor.
Step 5.2.3.1.2.3
Rewrite the expression.
Step 5.2.3.2
Move the negative in front of the fraction.
Step 6
Replace with .
Step 7
Set the numerator equal to zero.
Step 8
Step 8.1
Simplify .
Step 8.1.1
Raising to any positive power yields .
Step 8.1.2
Add and .
Step 8.2
Subtract from both sides of the equation.
Step 8.3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 8.4
Rewrite as .
Step 8.5
The complete solution is the result of both the positive and negative portions of the solution.
Step 8.5.1
First, use the positive value of the to find the first solution.
Step 8.5.2
Next, use the negative value of the to find the second solution.
Step 8.5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 9
Find the points where .
Step 10