# Calculus Examples

Step 1

Find the first derivative.

Differentiate.

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

Differentiate using the Power Rule which states that is where .

Evaluate .

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

Differentiate using the Power Rule which states that is where .

Multiply by .

Evaluate .

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

Differentiate using the Power Rule which states that is where .

Multiply by .

The first derivative of with respect to is .

Step 2

Set the first derivative equal to .

Factor the left side of the equation.

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor out of .

Factor.

Factor using the rational roots test.

If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.

Find every combination of . These are the possible roots of the polynomial function.

Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.

Substitute into the polynomial.

Raise to the power of .

Add and .

Subtract from .

Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

Divide by .

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .

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Divide the highest order term in the dividend by the highest order term in divisor .

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Multiply the new quotient term by the divisor.

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The expression needs to be subtracted from the dividend, so change all the signs in

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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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Pull the next terms from the original dividend down into the current dividend.

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Divide the highest order term in the dividend by the highest order term in divisor .

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Multiply the new quotient term by the divisor.

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The expression needs to be subtracted from the dividend, so change all the signs in

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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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Pull the next terms from the original dividend down into the current dividend.

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Divide the highest order term in the dividend by the highest order term in divisor .

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Multiply the new quotient term by the divisor.

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The expression needs to be subtracted from the dividend, so change all the signs in

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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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Since the remander is , the final answer is the quotient.

Write as a set of factors.

Remove unnecessary parentheses.

If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .

Set equal to and solve for .

Set equal to .

Add to both sides of the equation.

Set equal to and solve for .

Set equal to .

Solve for .

Use the quadratic formula to find the solutions.

Substitute the values , , and into the quadratic formula and solve for .

Simplify.

Simplify the numerator.

One to any power is one.

Multiply .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Multiply by .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

One to any power is one.

Multiply .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Multiply by .

Change the to .

Rewrite as .

Factor out of .

Factor out of .

Move the negative in front of the fraction.

Simplify the expression to solve for the portion of the .

Simplify the numerator.

One to any power is one.

Multiply .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Multiply by .

Change the to .

Rewrite as .

Factor out of .

Factor out of .

Move the negative in front of the fraction.

The final answer is the combination of both solutions.

The final solution is all the values that make true.

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

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Raising to any positive power yields .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Add and .

Subtract from .

The final answer is .

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

Decreasing on since

Decreasing on since

Step 6

Replace the variable with in the expression.

Simplify the result.

Simplify each term.

Raise to the power of .

Multiply by .

Multiply by .

Simplify by adding and subtracting.

Add and .

Subtract from .

The final answer is .

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