# Calculus Examples

Step 1

Step 1.1

Find the first derivative.

Step 1.1.1

Differentiate.

Step 1.1.1.1

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

Step 1.1.1.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2

Evaluate .

Step 1.1.2.1

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

Step 1.1.2.2

Differentiate using the Power Rule which states that is where .

Step 1.1.2.3

Multiply by .

Step 1.1.3

Evaluate .

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

Multiply by .

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 the left side of the equation.

Step 2.2.1

Factor out of .

Step 2.2.1.1

Factor out of .

Step 2.2.1.2

Factor out of .

Step 2.2.1.3

Factor out of .

Step 2.2.1.4

Factor out of .

Step 2.2.1.5

Factor out of .

Step 2.2.2

Factor.

Step 2.2.2.1

Factor using the rational roots test.

Step 2.2.2.1.1

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.

Step 2.2.2.1.2

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

Step 2.2.2.1.3

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

Step 2.2.2.1.3.1

Substitute into the polynomial.

Step 2.2.2.1.3.2

Raise to the power of .

Step 2.2.2.1.3.3

Add and .

Step 2.2.2.1.3.4

Subtract from .

Step 2.2.2.1.4

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.

Step 2.2.2.1.5

Divide by .

Step 2.2.2.1.5.1

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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Step 2.2.2.1.5.2

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

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Step 2.2.2.1.5.3

Multiply the new quotient term by the divisor.

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Step 2.2.2.1.5.4

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

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Step 2.2.2.1.5.5

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

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Step 2.2.2.1.5.6

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

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Step 2.2.2.1.5.7

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

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Step 2.2.2.1.5.8

Multiply the new quotient term by the divisor.

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Step 2.2.2.1.5.9

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

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Step 2.2.2.1.5.10

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

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Step 2.2.2.1.5.11

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

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Step 2.2.2.1.5.12

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

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Step 2.2.2.1.5.13

Multiply the new quotient term by the divisor.

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Step 2.2.2.1.5.14

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

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Step 2.2.2.1.5.15

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

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Step 2.2.2.1.5.16

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

Step 2.2.2.1.6

Write as a set of factors.

Step 2.2.2.2

Remove unnecessary parentheses.

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

Add to both sides of the equation.

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

Use the quadratic formula to find the solutions.

Step 2.5.2.2

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

Step 2.5.2.3

Simplify.

Step 2.5.2.3.1

Simplify the numerator.

Step 2.5.2.3.1.1

One to any power is one.

Step 2.5.2.3.1.2

Multiply .

Step 2.5.2.3.1.2.1

Multiply by .

Step 2.5.2.3.1.2.2

Multiply by .

Step 2.5.2.3.1.3

Subtract from .

Step 2.5.2.3.1.4

Rewrite as .

Step 2.5.2.3.1.5

Rewrite as .

Step 2.5.2.3.1.6

Rewrite as .

Step 2.5.2.3.2

Multiply by .

Step 2.5.2.4

Simplify the expression to solve for the portion of the .

Step 2.5.2.4.1

Simplify the numerator.

Step 2.5.2.4.1.1

One to any power is one.

Step 2.5.2.4.1.2

Multiply .

Step 2.5.2.4.1.2.1

Multiply by .

Step 2.5.2.4.1.2.2

Multiply by .

Step 2.5.2.4.1.3

Subtract from .

Step 2.5.2.4.1.4

Rewrite as .

Step 2.5.2.4.1.5

Rewrite as .

Step 2.5.2.4.1.6

Rewrite as .

Step 2.5.2.4.2

Multiply by .

Step 2.5.2.4.3

Change the to .

Step 2.5.2.4.4

Rewrite as .

Step 2.5.2.4.5

Factor out of .

Step 2.5.2.4.6

Factor out of .

Step 2.5.2.4.7

Move the negative in front of the fraction.

Step 2.5.2.5

Simplify the expression to solve for the portion of the .

Step 2.5.2.5.1

Simplify the numerator.

Step 2.5.2.5.1.1

One to any power is one.

Step 2.5.2.5.1.2

Multiply .

Step 2.5.2.5.1.2.1

Multiply by .

Step 2.5.2.5.1.2.2

Multiply by .

Step 2.5.2.5.1.3

Subtract from .

Step 2.5.2.5.1.4

Rewrite as .

Step 2.5.2.5.1.5

Rewrite as .

Step 2.5.2.5.1.6

Rewrite as .

Step 2.5.2.5.2

Multiply by .

Step 2.5.2.5.3

Change the to .

Step 2.5.2.5.4

Rewrite as .

Step 2.5.2.5.5

Factor out of .

Step 2.5.2.5.6

Factor out of .

Step 2.5.2.5.7

Move the negative in front of the fraction.

Step 2.5.2.6

The final answer is the combination of both solutions.

Step 2.6

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

Step 5.1

Replace the variable with in the expression.

Step 5.2

Simplify the result.

Step 5.2.1

Simplify each term.

Step 5.2.1.1

Raising to any positive power yields .

Step 5.2.1.2

Multiply by .

Step 5.2.1.3

Multiply by .

Step 5.2.2

Simplify by adding and subtracting.

Step 5.2.2.1

Add and .

Step 5.2.2.2

Subtract from .

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 each term.

Step 6.2.1.1

Raise to the power of .

Step 6.2.1.2

Multiply by .

Step 6.2.1.3

Multiply by .

Step 6.2.2

Simplify by adding and subtracting.

Step 6.2.2.1

Add and .

Step 6.2.2.2

Subtract from .

Step 6.2.3

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