# Calculus Examples

Differentiate both sides of the equation.

The derivative of with respect to is .

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 .

Reorder terms.

Reform the equation by setting the left side equal to the right side.

Replace with .

Rewrite the equation as .

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.

Simplify each term.

Raise to the power of .

Multiply by .

Multiply by .

Add and .

Add and .

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 .

Write as a set of factors.

Set equal to and solve for .

Set the factor equal to .

Subtract from both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Reduce the expression by cancelling the common factors.

Cancel the common factor.

Divide by .

Move the negative in front of the fraction.

Set equal to and solve for .

Set the factor equal to .

Use the quadratic formula to find the solutions.

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

Simplify.

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Rewrite as .

Pull terms out from under the radical, assuming positive real numbers.

Move to the left of .

Multiply by .

Simplify .

Simplify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Rewrite as .

Pull terms out from under the radical, assuming positive real numbers.

Move to the left of .

Multiply by .

Simplify .

Change the to .

Split the fraction into two fractions.

Simplify the expression to solve for the portion of the .

Simplify the numerator.

Raise to the power of .

Multiply by .

Multiply by .

Subtract from .

Rewrite as .

Rewrite as .

Rewrite as .

Rewrite as .

Pull terms out from under the radical, assuming positive real numbers.

Move to the left of .

Multiply by .

Simplify .

Change the to .

Split the fraction into two fractions.

Move the negative in front of the fraction.

The final answer is the combination of both solutions.

The solution is the result of and .

Remove the parentheses around the expression .

Simplify .

Simplify each term.

Use the power rule to distribute the exponent.

Apply the product rule to .

Apply the product rule to .

Multiply by by adding the exponents.

Move .

Multiply by .

Raise to the power of .

Use the power rule to combine exponents.

Add and .

Raise to the power of .

One to any power is one.

Raise to the power of .

Use the power rule to distribute the exponent.

Apply the product rule to .

Apply the product rule to .

Raise to the power of .

Multiply by .

One to any power is one.

Raise to the power of .

Cancel the common factor of .

Write as a fraction with denominator .

Factor out the greatest common factor .

Cancel the common factor.

Rewrite the expression.

Multiply and .

To write as a fraction with a common denominator, multiply by .

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

Combine.

Multiply by .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

Move the negative in front of the fraction.

Combine the numerators over the common denominator.

Add and .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Move the negative in front of the fraction.

Calculated values cannot contain imaginary components.

is not a valid value for x

Calculated values cannot contain imaginary components.

is not a valid value for x

Find the points where .