Calculus Examples

Find the Derivative - d/du (8u^2)/((u^2+u)^3)
Step 1
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Differentiate using the Power Rule.
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Step 3.1
Multiply the exponents in .
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Step 3.1.1
Apply the power rule and multiply exponents, .
Step 3.1.2
Multiply by .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Move to the left of .
Step 4
Differentiate using the chain rule, which states that is where and .
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Step 4.1
To apply the Chain Rule, set as .
Step 4.2
Differentiate using the Power Rule which states that is where .
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate.
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Step 5.1
Multiply by .
Step 5.2
By the Sum Rule, the derivative of with respect to is .
Step 5.3
Differentiate using the Power Rule which states that is where .
Step 5.4
Differentiate using the Power Rule which states that is where .
Step 5.5
Combine and .
Step 6
Simplify.
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Step 6.1
Apply the distributive property.
Step 6.2
Simplify the numerator.
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Step 6.2.1
Use the Binomial Theorem.
Step 6.2.2
Simplify each term.
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Step 6.2.2.1
Multiply the exponents in .
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Step 6.2.2.1.1
Apply the power rule and multiply exponents, .
Step 6.2.2.1.2
Multiply by .
Step 6.2.2.2
Multiply the exponents in .
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Step 6.2.2.2.1
Apply the power rule and multiply exponents, .
Step 6.2.2.2.2
Multiply by .
Step 6.2.2.3
Multiply by by adding the exponents.
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Step 6.2.2.3.1
Move .
Step 6.2.2.3.2
Multiply by .
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Step 6.2.2.3.2.1
Raise to the power of .
Step 6.2.2.3.2.2
Use the power rule to combine exponents.
Step 6.2.2.3.3
Add and .
Step 6.2.2.4
Multiply by by adding the exponents.
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Step 6.2.2.4.1
Move .
Step 6.2.2.4.2
Use the power rule to combine exponents.
Step 6.2.2.4.3
Add and .
Step 6.2.3
Apply the distributive property.
Step 6.2.4
Simplify.
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Step 6.2.4.1
Multiply by .
Step 6.2.4.2
Multiply by .
Step 6.2.5
Apply the distributive property.
Step 6.2.6
Simplify.
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Step 6.2.6.1
Multiply by by adding the exponents.
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Step 6.2.6.1.1
Move .
Step 6.2.6.1.2
Multiply by .
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Step 6.2.6.1.2.1
Raise to the power of .
Step 6.2.6.1.2.2
Use the power rule to combine exponents.
Step 6.2.6.1.3
Add and .
Step 6.2.6.2
Multiply by by adding the exponents.
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Step 6.2.6.2.1
Move .
Step 6.2.6.2.2
Multiply by .
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Step 6.2.6.2.2.1
Raise to the power of .
Step 6.2.6.2.2.2
Use the power rule to combine exponents.
Step 6.2.6.2.3
Add and .
Step 6.2.6.3
Multiply by by adding the exponents.
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Step 6.2.6.3.1
Move .
Step 6.2.6.3.2
Multiply by .
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Step 6.2.6.3.2.1
Raise to the power of .
Step 6.2.6.3.2.2
Use the power rule to combine exponents.
Step 6.2.6.3.3
Add and .
Step 6.2.6.4
Multiply by by adding the exponents.
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Step 6.2.6.4.1
Move .
Step 6.2.6.4.2
Multiply by .
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Step 6.2.6.4.2.1
Raise to the power of .
Step 6.2.6.4.2.2
Use the power rule to combine exponents.
Step 6.2.6.4.3
Add and .
Step 6.2.7
Apply the distributive property.
Step 6.2.8
Simplify.
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Step 6.2.8.1
Multiply by .
Step 6.2.8.2
Multiply by .
Step 6.2.8.3
Multiply by .
Step 6.2.8.4
Multiply by .
Step 6.2.9
Rewrite as .
Step 6.2.10
Expand using the FOIL Method.
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Step 6.2.10.1
Apply the distributive property.
Step 6.2.10.2
Apply the distributive property.
Step 6.2.10.3
Apply the distributive property.
Step 6.2.11
Simplify and combine like terms.
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Step 6.2.11.1
Simplify each term.
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Step 6.2.11.1.1
Multiply by by adding the exponents.
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Step 6.2.11.1.1.1
Use the power rule to combine exponents.
Step 6.2.11.1.1.2
Add and .
Step 6.2.11.1.2
Multiply by by adding the exponents.
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Step 6.2.11.1.2.1
Multiply by .
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Step 6.2.11.1.2.1.1
Raise to the power of .
Step 6.2.11.1.2.1.2
Use the power rule to combine exponents.
Step 6.2.11.1.2.2
Add and .
Step 6.2.11.1.3
Multiply by by adding the exponents.
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Step 6.2.11.1.3.1
Multiply by .
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Step 6.2.11.1.3.1.1
Raise to the power of .
Step 6.2.11.1.3.1.2
Use the power rule to combine exponents.
Step 6.2.11.1.3.2
Add and .
Step 6.2.11.1.4
Multiply by .
Step 6.2.11.2
Add and .
Step 6.2.12
Expand by multiplying each term in the first expression by each term in the second expression.
Step 6.2.13
Simplify each term.
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Step 6.2.13.1
Rewrite using the commutative property of multiplication.
Step 6.2.13.2
Multiply by by adding the exponents.
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Step 6.2.13.2.1
Move .
Step 6.2.13.2.2
Multiply by .
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Step 6.2.13.2.2.1
Raise to the power of .
Step 6.2.13.2.2.2
Use the power rule to combine exponents.
Step 6.2.13.2.3
Add and .
Step 6.2.13.3
Multiply by .
Step 6.2.13.4
Rewrite using the commutative property of multiplication.
Step 6.2.13.5
Multiply by by adding the exponents.
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Step 6.2.13.5.1
Move .
Step 6.2.13.5.2
Multiply by .
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Step 6.2.13.5.2.1
Raise to the power of .
Step 6.2.13.5.2.2
Use the power rule to combine exponents.
Step 6.2.13.5.3
Add and .
Step 6.2.13.6
Multiply by .
Step 6.2.13.7
Multiply by .
Step 6.2.13.8
Rewrite using the commutative property of multiplication.
Step 6.2.13.9
Multiply by by adding the exponents.
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Step 6.2.13.9.1
Move .
Step 6.2.13.9.2
Multiply by .
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Step 6.2.13.9.2.1
Raise to the power of .
Step 6.2.13.9.2.2
Use the power rule to combine exponents.
Step 6.2.13.9.3
Add and .
Step 6.2.13.10
Multiply by .
Step 6.2.14
Add and .
Step 6.2.15
Add and .
Step 6.2.16
Apply the distributive property.
Step 6.2.17
Simplify.
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Step 6.2.17.1
Rewrite using the commutative property of multiplication.
Step 6.2.17.2
Rewrite using the commutative property of multiplication.
Step 6.2.17.3
Rewrite using the commutative property of multiplication.
Step 6.2.17.4
Multiply by by adding the exponents.
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Step 6.2.17.4.1
Move .
Step 6.2.17.4.2
Use the power rule to combine exponents.
Step 6.2.17.4.3
Add and .
Step 6.2.18
Simplify each term.
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Step 6.2.18.1
Multiply by by adding the exponents.
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Step 6.2.18.1.1
Move .
Step 6.2.18.1.2
Use the power rule to combine exponents.
Step 6.2.18.1.3
Add and .
Step 6.2.18.2
Multiply by .
Step 6.2.18.3
Multiply by by adding the exponents.
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Step 6.2.18.3.1
Move .
Step 6.2.18.3.2
Use the power rule to combine exponents.
Step 6.2.18.3.3
Add and .
Step 6.2.18.4
Multiply by .
Step 6.2.18.5
Multiply by by adding the exponents.
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Step 6.2.18.5.1
Move .
Step 6.2.18.5.2
Use the power rule to combine exponents.
Step 6.2.18.5.3
Add and .
Step 6.2.18.6
Multiply by .
Step 6.2.19
Apply the distributive property.
Step 6.2.20
Simplify.
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Step 6.2.20.1
Multiply by .
Step 6.2.20.2
Multiply by .
Step 6.2.20.3
Multiply by .
Step 6.2.20.4
Multiply by .
Step 6.2.21
Subtract from .
Step 6.2.22
Subtract from .
Step 6.2.23
Subtract from .
Step 6.2.24
Subtract from .
Step 6.2.25
Rewrite in a factored form.
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Step 6.2.25.1
Factor out of .
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Step 6.2.25.1.1
Factor out of .
Step 6.2.25.1.2
Factor out of .
Step 6.2.25.1.3
Factor out of .
Step 6.2.25.1.4
Factor out of .
Step 6.2.25.1.5
Factor out of .
Step 6.2.25.1.6
Factor out of .
Step 6.2.25.1.7
Factor out of .
Step 6.2.25.2
Factor using the rational roots test.
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Step 6.2.25.2.1
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 6.2.25.2.2
Find every combination of . These are the possible roots of the polynomial function.
Step 6.2.25.2.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
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Step 6.2.25.2.3.1
Substitute into the polynomial.
Step 6.2.25.2.3.2
Raise to the power of .
Step 6.2.25.2.3.3
Multiply by .
Step 6.2.25.2.3.4
Raise to the power of .
Step 6.2.25.2.3.5
Multiply by .
Step 6.2.25.2.3.6
Subtract from .
Step 6.2.25.2.3.7
Multiply by .
Step 6.2.25.2.3.8
Add and .
Step 6.2.25.2.3.9
Subtract from .
Step 6.2.25.2.4
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 6.2.25.2.5
Divide by .
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Step 6.2.25.2.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
+----
Step 6.2.25.2.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
-
+----
Step 6.2.25.2.5.3
Multiply the new quotient term by the divisor.
-
+----
--
Step 6.2.25.2.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
-
+----
++
Step 6.2.25.2.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
-
+----
++
-
Step 6.2.25.2.5.6
Pull the next terms from the original dividend down into the current dividend.
-
+----
++
--
Step 6.2.25.2.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
--
+----
++
--
Step 6.2.25.2.5.8
Multiply the new quotient term by the divisor.
--
+----
++
--
--
Step 6.2.25.2.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
--
+----
++
--
++
Step 6.2.25.2.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
--
+----
++
--
++
-
Step 6.2.25.2.5.11
Pull the next terms from the original dividend down into the current dividend.
--
+----
++
--
++
--
Step 6.2.25.2.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
---
+----
++
--
++
--
Step 6.2.25.2.5.13
Multiply the new quotient term by the divisor.
---
+----
++
--
++
--
--
Step 6.2.25.2.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
---
+----
++
--
++
--
++
Step 6.2.25.2.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
---
+----
++
--
++
--
++
Step 6.2.25.2.5.16
Since the remander is , the final answer is the quotient.
Step 6.2.25.2.6
Write as a set of factors.
Step 6.2.25.3
Factor by grouping.
Step 6.2.25.4
Combine exponents.
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Step 6.2.25.4.1
Raise to the power of .
Step 6.2.25.4.2
Raise to the power of .
Step 6.2.25.4.3
Use the power rule to combine exponents.
Step 6.2.25.4.4
Add and .
Step 6.3
Simplify the denominator.
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Step 6.3.1
Factor out of .
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Step 6.3.1.1
Factor out of .
Step 6.3.1.2
Raise to the power of .
Step 6.3.1.3
Factor out of .
Step 6.3.1.4
Factor out of .
Step 6.3.2
Apply the product rule to .
Step 6.4
Cancel the common factor of and .
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Step 6.4.1
Factor out of .
Step 6.4.2
Cancel the common factors.
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Step 6.4.2.1
Factor out of .
Step 6.4.2.2
Cancel the common factor.
Step 6.4.2.3
Rewrite the expression.
Step 6.5
Cancel the common factor of and .
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Step 6.5.1
Factor out of .
Step 6.5.2
Cancel the common factors.
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Step 6.5.2.1
Factor out of .
Step 6.5.2.2
Cancel the common factor.
Step 6.5.2.3
Rewrite the expression.
Step 6.6
Factor out of .
Step 6.7
Rewrite as .
Step 6.8
Factor out of .
Step 6.9
Rewrite as .
Step 6.10
Move the negative in front of the fraction.