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Calculus Examples
Step 1
Step 1.1
Find the first derivative.
Step 1.1.1
Use to rewrite as .
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 Power Rule which states that is where .
Step 1.1.2.3
Replace all occurrences of with .
Step 1.1.3
To write as a fraction with a common denominator, multiply by .
Step 1.1.4
Combine and .
Step 1.1.5
Combine the numerators over the common denominator.
Step 1.1.6
Simplify the numerator.
Step 1.1.6.1
Multiply by .
Step 1.1.6.2
Subtract from .
Step 1.1.7
Combine fractions.
Step 1.1.7.1
Move the negative in front of the fraction.
Step 1.1.7.2
Combine and .
Step 1.1.7.3
Move to the denominator using the negative exponent rule .
Step 1.1.8
By the Sum Rule, the derivative of with respect to is .
Step 1.1.9
Differentiate using the Power Rule which states that is where .
Step 1.1.10
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.11
Simplify terms.
Step 1.1.11.1
Add and .
Step 1.1.11.2
Combine and .
Step 1.1.11.3
Combine and .
Step 1.1.11.4
Cancel the common factor.
Step 1.1.11.5
Rewrite the expression.
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
Set the numerator equal to zero.
Step 3
The values which make the derivative equal to are .
Step 4
After finding the point that makes the derivative equal to or undefined, the interval to check where is increasing and where it is decreasing is .
Step 5
Step 5.1
Replace the variable with in the expression.
Step 5.2
Simplify the result.
Step 5.2.1
Simplify the denominator.
Step 5.2.1.1
Raise to the power of .
Step 5.2.1.2
Add and .
Step 5.2.2
Move the negative in front of the fraction.
Step 5.2.3
The final answer is .
Step 5.3
At the derivative is . Since this is negative, the function is decreasing on .
Decreasing on since
Decreasing on since
Step 6
Step 6.1
Replace the variable with in the expression.
Step 6.2
Simplify the result.
Step 6.2.1
Simplify the denominator.
Step 6.2.1.1
One to any power is one.
Step 6.2.1.2
Add and .
Step 6.2.2
The final answer is .
Step 6.3
At the derivative is . Since this is positive, the function is increasing on .
Increasing on since
Increasing on since
Step 7
List the intervals on which the function is increasing and decreasing.
Increasing on:
Decreasing on:
Step 8