Calculus Examples

Find dy/dx y=((x^2+1)^10(x^2+2)^3)/((e^x+1)^7)
Step 1
Differentiate both sides of the equation.
Step 2
The derivative of with respect to is .
Step 3
Differentiate the right side of the equation.
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Step 3.1
Differentiate using the Quotient Rule which states that is where and .
Step 3.2
Multiply the exponents in .
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Step 3.2.1
Apply the power rule and multiply exponents, .
Step 3.2.2
Multiply by .
Step 3.3
Differentiate using the Product Rule which states that is where and .
Step 3.4
Differentiate using the chain rule, which states that is where and .
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Step 3.4.1
To apply the Chain Rule, set as .
Step 3.4.2
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Differentiate.
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Step 3.5.1
Move to the left of .
Step 3.5.2
By the Sum Rule, the derivative of with respect to is .
Step 3.5.3
Differentiate using the Power Rule which states that is where .
Step 3.5.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.5
Simplify the expression.
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Step 3.5.5.1
Add and .
Step 3.5.5.2
Multiply by .
Step 3.6
Differentiate using the chain rule, which states that is where and .
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Step 3.6.1
To apply the Chain Rule, set as .
Step 3.6.2
Differentiate using the Power Rule which states that is where .
Step 3.6.3
Replace all occurrences of with .
Step 3.7
Differentiate.
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Step 3.7.1
Move to the left of .
Step 3.7.2
By the Sum Rule, the derivative of with respect to is .
Step 3.7.3
Differentiate using the Power Rule which states that is where .
Step 3.7.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.7.5
Simplify the expression.
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Step 3.7.5.1
Add and .
Step 3.7.5.2
Multiply by .
Step 3.8
Differentiate using the chain rule, which states that is where and .
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Step 3.8.1
To apply the Chain Rule, set as .
Step 3.8.2
Differentiate using the Power Rule which states that is where .
Step 3.8.3
Replace all occurrences of with .
Step 3.9
Simplify with factoring out.
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Step 3.9.1
Multiply by .
Step 3.9.2
Factor out of .
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Step 3.9.2.1
Factor out of .
Step 3.9.2.2
Factor out of .
Step 3.9.2.3
Factor out of .
Step 3.10
Cancel the common factors.
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Step 3.10.1
Factor out of .
Step 3.10.2
Cancel the common factor.
Step 3.10.3
Rewrite the expression.
Step 3.11
By the Sum Rule, the derivative of with respect to is .
Step 3.12
Differentiate using the Exponential Rule which states that is where =.
Step 3.13
Differentiate using the Constant Rule.
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Step 3.13.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.13.2
Add and .
Step 3.14
Simplify the numerator.
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Step 3.14.1
Simplify each term.
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Step 3.14.1.1
Rewrite as .
Step 3.14.1.2
Expand using the FOIL Method.
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Step 3.14.1.2.1
Apply the distributive property.
Step 3.14.1.2.2
Apply the distributive property.
Step 3.14.1.2.3
Apply the distributive property.
Step 3.14.1.3
Simplify and combine like terms.
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Step 3.14.1.3.1
Simplify each term.
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Step 3.14.1.3.1.1
Multiply by by adding the exponents.
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Step 3.14.1.3.1.1.1
Use the power rule to combine exponents.
Step 3.14.1.3.1.1.2
Add and .
Step 3.14.1.3.1.2
Move to the left of .
Step 3.14.1.3.1.3
Multiply by .
Step 3.14.1.3.2
Add and .
Step 3.14.1.4
Apply the distributive property.
Step 3.14.1.5
Simplify.
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Step 3.14.1.5.1
Rewrite using the commutative property of multiplication.
Step 3.14.1.5.2
Multiply by .
Step 3.14.1.6
Multiply by .
Step 3.14.1.7
Apply the distributive property.
Step 3.14.1.8
Simplify.
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Step 3.14.1.8.1
Multiply by by adding the exponents.
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Step 3.14.1.8.1.1
Move .
Step 3.14.1.8.1.2
Multiply by .
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Step 3.14.1.8.1.2.1
Raise to the power of .
Step 3.14.1.8.1.2.2
Use the power rule to combine exponents.
Step 3.14.1.8.1.3
Add and .
Step 3.14.1.8.2
Multiply by by adding the exponents.
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Step 3.14.1.8.2.1
Move .
Step 3.14.1.8.2.2
Multiply by .
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Step 3.14.1.8.2.2.1
Raise to the power of .
Step 3.14.1.8.2.2.2
Use the power rule to combine exponents.
Step 3.14.1.8.2.3
Add and .
Step 3.14.1.9
Use the Binomial Theorem.
Step 3.14.1.10
Simplify each term.
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Step 3.14.1.10.1
Multiply the exponents in .
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Step 3.14.1.10.1.1
Apply the power rule and multiply exponents, .
Step 3.14.1.10.1.2
Multiply by .
Step 3.14.1.10.2
Multiply the exponents in .
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Step 3.14.1.10.2.1
Apply the power rule and multiply exponents, .
Step 3.14.1.10.2.2
Multiply by .
Step 3.14.1.10.3
Multiply by .
Step 3.14.1.10.4
Raise to the power of .
Step 3.14.1.10.5
Multiply by .
Step 3.14.1.10.6
Raise to the power of .
Step 3.14.1.11
Apply the distributive property.
Step 3.14.1.12
Simplify.
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Step 3.14.1.12.1
Multiply by .
Step 3.14.1.12.2
Multiply by .
Step 3.14.1.12.3
Multiply by .
Step 3.14.1.13
Apply the distributive property.
Step 3.14.1.14
Apply the distributive property.
Step 3.14.1.15
Simplify.
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Step 3.14.1.15.1
Multiply by by adding the exponents.
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Step 3.14.1.15.1.1
Move .
Step 3.14.1.15.1.2
Multiply by .
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Step 3.14.1.15.1.2.1
Raise to the power of .
Step 3.14.1.15.1.2.2
Use the power rule to combine exponents.
Step 3.14.1.15.1.3
Add and .
Step 3.14.1.15.2
Multiply by by adding the exponents.
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Step 3.14.1.15.2.1
Move .
Step 3.14.1.15.2.2
Multiply by .
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Step 3.14.1.15.2.2.1
Raise to the power of .
Step 3.14.1.15.2.2.2
Use the power rule to combine exponents.
Step 3.14.1.15.2.3
Add and .
Step 3.14.1.15.3
Multiply by by adding the exponents.
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Step 3.14.1.15.3.1
Move .
Step 3.14.1.15.3.2
Multiply by .
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Step 3.14.1.15.3.2.1
Raise to the power of .
Step 3.14.1.15.3.2.2
Use the power rule to combine exponents.
Step 3.14.1.15.3.3
Add and .
Step 3.14.2
Expand by multiplying each term in the first expression by each term in the second expression.
Step 3.14.3
Simplify each term.
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Step 3.14.3.1
Rewrite using the commutative property of multiplication.
Step 3.14.3.2
Rewrite using the commutative property of multiplication.
Step 3.14.3.3
Rewrite using the commutative property of multiplication.
Step 3.14.3.4
Rewrite using the commutative property of multiplication.
Step 3.14.3.5
Rewrite using the commutative property of multiplication.
Step 3.14.3.6
Rewrite using the commutative property of multiplication.
Step 3.14.3.7
Rewrite using the commutative property of multiplication.
Step 3.14.3.8
Multiply by .
Step 3.14.3.9
Multiply by .
Step 3.14.3.10
Multiply by .
Step 3.14.3.11
Multiply by .
Step 3.14.3.12
Multiply by .
Step 3.14.3.13
Multiply by .
Step 3.14.3.14
Multiply by .
Step 3.14.4
Use the Binomial Theorem.
Step 3.14.5
Simplify each term.
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Step 3.14.5.1
Multiply the exponents in .
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Step 3.14.5.1.1
Apply the power rule and multiply exponents, .
Step 3.14.5.1.2
Multiply by .
Step 3.14.5.2
Multiply the exponents in .
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Step 3.14.5.2.1
Apply the power rule and multiply exponents, .
Step 3.14.5.2.2
Multiply by .
Step 3.14.5.3
Multiply by .
Step 3.14.5.4
Raise to the power of .
Step 3.14.5.5
Multiply by .
Step 3.14.5.6
Raise to the power of .
Step 3.14.6
Apply the distributive property.
Step 3.14.7
Simplify.
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Step 3.14.7.1
Rewrite using the commutative property of multiplication.
Step 3.14.7.2
Rewrite using the commutative property of multiplication.
Step 3.14.7.3
Multiply by .
Step 3.14.8
Simplify each term.
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Step 3.14.8.1
Multiply by .
Step 3.14.8.2
Multiply by .
Step 3.14.9
Apply the distributive property.
Step 3.14.10
Rewrite in a factored form.
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Step 3.14.10.1
Factor out of .
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Step 3.14.10.1.1
Factor out of .
Step 3.14.10.1.2
Factor out of .
Step 3.14.10.1.3
Factor out of .
Step 3.14.10.1.4
Factor out of .
Step 3.14.10.1.5
Factor out of .
Step 3.14.10.1.6
Factor out of .
Step 3.14.10.1.7
Factor out of .
Step 3.14.10.1.8
Factor out of .
Step 3.14.10.1.9
Factor out of .
Step 3.14.10.1.10
Factor out of .
Step 3.14.10.1.11
Factor out of .
Step 3.14.10.1.12
Factor out of .
Step 3.14.10.1.13
Factor out of .
Step 3.14.10.1.14
Factor out of .
Step 3.14.10.1.15
Factor out of .
Step 3.14.10.1.16
Factor out of .
Step 3.14.10.1.17
Factor out of .
Step 3.14.10.1.18
Factor out of .
Step 3.14.10.1.19
Factor out of .
Step 3.14.10.1.20
Factor out of .
Step 3.14.10.1.21
Factor out of .
Step 3.14.10.1.22
Factor out of .
Step 3.14.10.1.23
Factor out of .
Step 3.14.10.1.24
Factor out of .
Step 3.14.10.1.25
Factor out of .
Step 3.14.10.1.26
Factor out of .
Step 3.14.10.1.27
Factor out of .
Step 3.14.10.1.28
Factor out of .
Step 3.14.10.1.29
Factor out of .
Step 3.14.10.1.30
Factor out of .
Step 3.14.10.1.31
Factor out of .
Step 3.14.10.1.32
Factor out of .
Step 3.14.10.1.33
Factor out of .
Step 3.14.10.1.34
Factor out of .
Step 3.14.10.1.35
Factor out of .
Step 3.14.10.2
Apply the distributive property.
Step 3.14.10.3
Multiply by by adding the exponents.
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Step 3.14.10.3.1
Use the power rule to combine exponents.
Step 3.14.10.3.2
Add and .
Step 3.14.10.4
Multiply by .
Step 3.14.10.5
Apply the distributive property.
Step 3.14.10.6
Apply the distributive property.
Step 3.14.10.7
Multiply by by adding the exponents.
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Step 3.14.10.7.1
Use the power rule to combine exponents.
Step 3.14.10.7.2
Add and .
Step 3.14.10.8
Multiply by .
Step 3.14.10.9
Apply the distributive property.
Step 3.14.10.10
Apply the distributive property.
Step 3.14.10.11
Multiply by by adding the exponents.
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Step 3.14.10.11.1
Multiply by .
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Step 3.14.10.11.1.1
Raise to the power of .
Step 3.14.10.11.1.2
Use the power rule to combine exponents.
Step 3.14.10.11.2
Add and .
Step 3.14.10.12
Multiply by .
Step 3.14.10.13
Apply the distributive property.
Step 3.14.10.14
Apply the distributive property.
Step 3.14.10.15
Multiply by .
Step 3.14.10.16
Apply the distributive property.
Step 3.14.10.17
Multiply by by adding the exponents.
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Step 3.14.10.17.1
Move .
Step 3.14.10.17.2
Use the power rule to combine exponents.
Step 3.14.10.17.3
Add and .
Step 3.14.10.18
Apply the distributive property.
Step 3.14.10.19
Multiply by .
Step 3.14.10.20
Apply the distributive property.
Step 3.14.10.21
Multiply by by adding the exponents.
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Step 3.14.10.21.1
Move .
Step 3.14.10.21.2
Use the power rule to combine exponents.
Step 3.14.10.21.3
Add and .
Step 3.14.10.22
Apply the distributive property.
Step 3.14.10.23
Multiply by .
Step 3.14.10.24
Apply the distributive property.
Step 3.14.10.25
Multiply by by adding the exponents.
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Step 3.14.10.25.1
Move .
Step 3.14.10.25.2
Multiply by .
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Step 3.14.10.25.2.1
Raise to the power of .
Step 3.14.10.25.2.2
Use the power rule to combine exponents.
Step 3.14.10.25.3
Add and .
Step 3.14.10.26
Apply the distributive property.
Step 3.14.10.27
Multiply by .
Step 3.14.10.28
Apply the distributive property.
Step 3.14.10.29
Multiply by by adding the exponents.
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Step 3.14.10.29.1
Move .
Step 3.14.10.29.2
Use the power rule to combine exponents.
Step 3.14.10.29.3
Add and .
Step 3.14.10.30
Apply the distributive property.
Step 3.14.10.31
Apply the distributive property.
Step 3.14.10.32
Multiply by .
Step 3.14.10.33
Apply the distributive property.
Step 3.14.10.34
Multiply by by adding the exponents.
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Step 3.14.10.34.1
Move .
Step 3.14.10.34.2
Use the power rule to combine exponents.
Step 3.14.10.34.3
Add and .
Step 3.14.10.35
Apply the distributive property.
Step 3.14.10.36
Apply the distributive property.
Step 3.14.10.37
Multiply by .
Step 3.14.10.38
Apply the distributive property.
Step 3.14.10.39
Multiply by by adding the exponents.
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Step 3.14.10.39.1
Move .
Step 3.14.10.39.2
Use the power rule to combine exponents.
Step 3.14.10.39.3
Add and .
Step 3.14.10.40
Apply the distributive property.
Step 3.14.10.41
Apply the distributive property.
Step 3.14.10.42
Multiply by .
Step 3.14.10.43
Apply the distributive property.
Step 3.14.10.44
Add and .
Step 3.14.10.45
Add and .
Step 3.14.10.46
Add and .
Step 3.14.10.47
Add and .
Step 3.14.10.48
Add and .
Step 3.14.10.49
Add and .
Step 3.14.10.50
Add and .
Step 3.14.10.51
Add and .
Step 3.14.10.52
Add and .
Step 3.14.10.53
Add and .
Step 3.14.10.54
Add and .
Step 3.14.10.55
Add and .
Step 3.14.10.56
Subtract from .
Step 3.14.10.57
Subtract from .
Step 3.14.10.58
Subtract from .
Step 3.14.10.59
Reorder terms.
Step 4
Reform the equation by setting the left side equal to the right side.
Step 5
Replace with .