Calculus Examples

Find Where Increasing/Decreasing Using Derivatives
Step 1
Find the first derivative.
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Find the first derivative.
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Differentiate.
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By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Evaluate .
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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 .
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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 then solve the equation .
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Set the first derivative equal to .
Factor the left side of the equation.
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Factor out of .
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Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor.
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Factor using the rational roots test.
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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.
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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 .
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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 .
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Set equal to .
Add to both sides of the equation.
Set equal to and solve for .
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Set equal to .
Solve for .
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Use the quadratic formula to find the solutions.
Substitute the values , , and into the quadratic formula and solve for .
Simplify.
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Simplify the numerator.
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One to any power is one.
Multiply .
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Multiply by .
Multiply by .
Subtract from .
Rewrite as .
Rewrite as .
Rewrite as .
Multiply by .
Simplify the expression to solve for the portion of the .
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Simplify the numerator.
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One to any power is one.
Multiply .
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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 .
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Simplify the numerator.
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One to any power is one.
Multiply .
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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
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raising to any positive power yields .
Multiply by .
Multiply by .
Simplify by adding and subtracting.
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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
Substitute a value from the interval into the derivative to determine if the function is increasing or decreasing.
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Replace the variable with in the expression.
Simplify the result.
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Simplify each term.
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Raise to the power of .
Multiply by .
Multiply by .
Simplify by adding and subtracting.
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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
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