Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

The derivative of with respect to is .

Replace all occurrences of with .

Differentiate.

By the Sum Rule, the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Simplify the expression.

Multiply by .

Reorder the factors of .

Set the derivative equal to .

Simplify the left side.

Multiply and .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Simplify each term.

Rewrite as .

Expand using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply by by adding the exponents.

Use the power rule to combine exponents.

Add and .

Rewrite using the commutative property of multiplication.

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Multiply by .

Multiply by .

Subtract from .

Reorder terms.

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

The LCM is the smallest number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for is itself.

occurs time.

The LCM of is the result of multiplying all factors the greatest number of times they occur in either term.

Multiply each term by and simplify.

Multiply each term in by in order to remove all the denominators from the equation.

Simplify the left side of the equation by cancelling the common factors.

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Apply the distributive property.

Multiply by .

Multiply by .

Solve the equation.

Add to both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Divide by .

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 .

Replace the variable with in the expression.

Simplify the result.

Simplify the denominator.

Simplify each term.

Raising to any positive power yields .

Multiply by .

Add and .

Raising to any positive power yields .

Add and .

Simplify the expression.

Multiply by .

Subtract from .

Divide by .

Multiply by .

The final answer is .

At the derivative is . Since this is negative, the function is decreasing on .

Decreasing on since

Decreasing on since

Replace the variable with in the expression.

Simplify the result.

Simplify the denominator.

Simplify each term.

Raise to the power of .

Multiply by .

Subtract from .

Raising to any positive power yields .

Add and .

Simplify the expression.

Multiply by .

Subtract from .

Divide by .

Multiply by .

The final answer is .

At the derivative is . Since this is positive, the function is increasing on .

Increasing on since

Increasing on since

List the intervals on which the function is increasing and decreasing.

Increasing on:

Decreasing on:

Find Where Increasing/Decreasing arctan(x^2-2x)