Calculus Examples

Find dy/dx y = natural log of (e^(x^2)(2x+1)(3x+1)^2)/( square root of 4x+1)
Step 1
Use to rewrite as .
Step 2
Differentiate both sides of the equation.
Step 3
The derivative of with respect to is .
Step 4
Differentiate the right side of the equation.
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Step 4.1
Differentiate using the chain rule, which states that is where and .
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Step 4.1.1
To apply the Chain Rule, set as .
Step 4.1.2
The derivative of with respect to is .
Step 4.1.3
Replace all occurrences of with .
Step 4.2
Multiply by the reciprocal of the fraction to divide by .
Step 4.3
Multiply by .
Step 4.4
Differentiate using the Quotient Rule which states that is where and .
Step 4.5
Multiply the exponents in .
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Step 4.5.1
Apply the power rule and multiply exponents, .
Step 4.5.2
Cancel the common factor of .
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Step 4.5.2.1
Cancel the common factor.
Step 4.5.2.2
Rewrite the expression.
Step 4.6
Simplify.
Step 4.7
Differentiate using the Product Rule which states that is where and .
Step 4.8
Differentiate using the chain rule, which states that is where and .
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Step 4.8.1
To apply the Chain Rule, set as .
Step 4.8.2
Differentiate using the Power Rule which states that is where .
Step 4.8.3
Replace all occurrences of with .
Step 4.9
Differentiate.
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Step 4.9.1
By the Sum Rule, the derivative of with respect to is .
Step 4.9.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.9.3
Differentiate using the Power Rule which states that is where .
Step 4.9.4
Multiply by .
Step 4.9.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.9.6
Simplify the expression.
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Step 4.9.6.1
Add and .
Step 4.9.6.2
Multiply by .
Step 4.10
Differentiate using the Product Rule which states that is where and .
Step 4.11
Differentiate.
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Step 4.11.1
By the Sum Rule, the derivative of with respect to is .
Step 4.11.2
Since is constant with respect to , the derivative of with respect to is .
Step 4.11.3
Differentiate using the Power Rule which states that is where .
Step 4.11.4
Multiply by .
Step 4.11.5
Since is constant with respect to , the derivative of with respect to is .
Step 4.11.6
Simplify the expression.
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Step 4.11.6.1
Add and .
Step 4.11.6.2
Move to the left of .
Step 4.12
Differentiate using the chain rule, which states that is where and .
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Step 4.12.1
To apply the Chain Rule, set as .
Step 4.12.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.12.3
Replace all occurrences of with .
Step 4.13
Differentiate using the Power Rule which states that is where .
Step 4.14
Differentiate using the chain rule, which states that is where and .
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Step 4.14.1
To apply the Chain Rule, set as .
Step 4.14.2
Differentiate using the Power Rule which states that is where .
Step 4.14.3
Replace all occurrences of with .
Step 4.15
To write as a fraction with a common denominator, multiply by .
Step 4.16
Combine and .
Step 4.17
Combine the numerators over the common denominator.
Step 4.18
Simplify the numerator.
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Step 4.18.1
Multiply by .
Step 4.18.2
Subtract from .
Step 4.19
Combine fractions.
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Step 4.19.1
Move the negative in front of the fraction.
Step 4.19.2
Combine and .
Step 4.19.3
Move to the denominator using the negative exponent rule .
Step 4.19.4
Combine and .
Step 4.19.5
Combine and .
Step 4.20
By the Sum Rule, the derivative of with respect to is .
Step 4.21
Since is constant with respect to , the derivative of with respect to is .
Step 4.22
Differentiate using the Power Rule which states that is where .
Step 4.23
Multiply by .
Step 4.24
Since is constant with respect to , the derivative of with respect to is .
Step 4.25
Simplify terms.
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Step 4.25.1
Add and .
Step 4.25.2
Multiply by .
Step 4.25.3
Combine and .
Step 4.25.4
Factor out of .
Step 4.26
Cancel the common factors.
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Step 4.26.1
Factor out of .
Step 4.26.2
Cancel the common factor.
Step 4.26.3
Rewrite the expression.
Step 4.27
Move the negative in front of the fraction.
Step 4.28
Multiply by .
Step 4.29
Move to the denominator using the negative exponent rule .
Step 4.30
Multiply by by adding the exponents.
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Step 4.30.1
Move .
Step 4.30.2
Multiply by .
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Step 4.30.2.1
Raise to the power of .
Step 4.30.2.2
Use the power rule to combine exponents.
Step 4.30.3
Write as a fraction with a common denominator.
Step 4.30.4
Combine the numerators over the common denominator.
Step 4.30.5
Add and .
Step 4.31
Simplify.
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Step 4.31.1
Apply the distributive property.
Step 4.31.2
Apply the distributive property.
Step 4.31.3
Apply the distributive property.
Step 4.31.4
Apply the distributive property.
Step 4.31.5
Apply the distributive property.
Step 4.31.6
Simplify the numerator.
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Step 4.31.6.1
Simplify each term.
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Step 4.31.6.1.1
Simplify each term.
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Step 4.31.6.1.1.1
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.1.2
Multiply by .
Step 4.31.6.1.2
Simplify each term.
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Step 4.31.6.1.2.1
Multiply by .
Step 4.31.6.1.2.2
Multiply by .
Step 4.31.6.1.3
Expand using the FOIL Method.
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Step 4.31.6.1.3.1
Apply the distributive property.
Step 4.31.6.1.3.2
Apply the distributive property.
Step 4.31.6.1.3.3
Apply the distributive property.
Step 4.31.6.1.4
Simplify and combine like terms.
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Step 4.31.6.1.4.1
Simplify each term.
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Step 4.31.6.1.4.1.1
Multiply by by adding the exponents.
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Step 4.31.6.1.4.1.1.1
Move .
Step 4.31.6.1.4.1.1.2
Multiply by .
Step 4.31.6.1.4.1.2
Multiply by .
Step 4.31.6.1.4.1.3
Multiply by .
Step 4.31.6.1.4.1.4
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.4.1.5
Move to the left of .
Step 4.31.6.1.4.2
Add and .
Step 4.31.6.1.5
Rewrite as .
Step 4.31.6.1.6
Expand using the FOIL Method.
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Step 4.31.6.1.6.1
Apply the distributive property.
Step 4.31.6.1.6.2
Apply the distributive property.
Step 4.31.6.1.6.3
Apply the distributive property.
Step 4.31.6.1.7
Simplify and combine like terms.
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Step 4.31.6.1.7.1
Simplify each term.
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Step 4.31.6.1.7.1.1
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.7.1.2
Multiply by by adding the exponents.
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Step 4.31.6.1.7.1.2.1
Move .
Step 4.31.6.1.7.1.2.2
Multiply by .
Step 4.31.6.1.7.1.3
Multiply by .
Step 4.31.6.1.7.1.4
Multiply by .
Step 4.31.6.1.7.1.5
Multiply by .
Step 4.31.6.1.7.1.6
Multiply by .
Step 4.31.6.1.7.2
Add and .
Step 4.31.6.1.8
Simplify each term.
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Step 4.31.6.1.8.1
Multiply by by adding the exponents.
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Step 4.31.6.1.8.1.1
Move .
Step 4.31.6.1.8.1.2
Multiply by .
Step 4.31.6.1.8.2
Multiply by .
Step 4.31.6.1.8.3
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.8.4
Multiply by .
Step 4.31.6.1.9
Expand by multiplying each term in the first expression by each term in the second expression.
Step 4.31.6.1.10
Simplify each term.
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Step 4.31.6.1.10.1
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.10.2
Multiply by .
Step 4.31.6.1.10.3
Multiply by by adding the exponents.
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Step 4.31.6.1.10.3.1
Move .
Step 4.31.6.1.10.3.2
Use the power rule to combine exponents.
Step 4.31.6.1.10.3.3
Add and .
Step 4.31.6.1.10.4
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.10.5
Multiply by .
Step 4.31.6.1.10.6
Multiply by by adding the exponents.
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Step 4.31.6.1.10.6.1
Move .
Step 4.31.6.1.10.6.2
Multiply by .
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Step 4.31.6.1.10.6.2.1
Raise to the power of .
Step 4.31.6.1.10.6.2.2
Use the power rule to combine exponents.
Step 4.31.6.1.10.6.3
Add and .
Step 4.31.6.1.10.7
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.10.8
Multiply by .
Step 4.31.6.1.10.9
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.10.10
Multiply by .
Step 4.31.6.1.10.11
Multiply by by adding the exponents.
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Step 4.31.6.1.10.11.1
Move .
Step 4.31.6.1.10.11.2
Multiply by .
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Step 4.31.6.1.10.11.2.1
Raise to the power of .
Step 4.31.6.1.10.11.2.2
Use the power rule to combine exponents.
Step 4.31.6.1.10.11.3
Add and .
Step 4.31.6.1.10.12
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.10.13
Multiply by .
Step 4.31.6.1.10.14
Multiply by by adding the exponents.
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Step 4.31.6.1.10.14.1
Move .
Step 4.31.6.1.10.14.2
Multiply by .
Step 4.31.6.1.10.15
Rewrite using the commutative property of multiplication.
Step 4.31.6.1.10.16
Multiply by .
Step 4.31.6.1.10.17
Multiply by .
Step 4.31.6.1.10.18
Multiply by .
Step 4.31.6.1.10.19
Multiply by .
Step 4.31.6.1.11
Add and .
Step 4.31.6.1.12
Add and .
Step 4.31.6.1.13
Add and .
Step 4.31.6.1.14
Add and .
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Step 4.31.6.1.14.1
Move .
Step 4.31.6.1.14.2
Add and .
Step 4.31.6.2
Add and .
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Step 4.31.6.2.1
Move .
Step 4.31.6.2.2
Add and .
Step 4.31.6.3
Add and .
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Step 4.31.6.3.1
Move .
Step 4.31.6.3.2
Add and .
Step 4.31.6.4
Add and .
Step 4.31.6.5
Apply the distributive property.
Step 4.31.6.6
Simplify.
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Step 4.31.6.6.1
Rewrite using the commutative property of multiplication.
Step 4.31.6.6.2
Rewrite using the commutative property of multiplication.
Step 4.31.6.6.3
Rewrite using the commutative property of multiplication.
Step 4.31.6.6.4
Rewrite using the commutative property of multiplication.
Step 4.31.6.6.5
Rewrite using the commutative property of multiplication.
Step 4.31.6.7
Apply the distributive property.
Step 4.31.6.8
Multiply .
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Step 4.31.6.8.1
Multiply by .
Step 4.31.6.8.2
Combine and .
Step 4.31.6.8.3
Multiply by .
Step 4.31.6.8.4
Combine and .
Step 4.31.6.9
Multiply by .
Step 4.31.6.10
Move the negative in front of the fraction.
Step 4.31.6.11
Combine the numerators over the common denominator.
Step 4.31.6.12
Factor out of .
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Step 4.31.6.12.1
Factor out of .
Step 4.31.6.12.2
Factor out of .
Step 4.31.6.12.3
Factor out of .
Step 4.31.6.13
To write as a fraction with a common denominator, multiply by .
Step 4.31.6.14
Combine the numerators over the common denominator.
Step 4.31.6.15
Move .
Step 4.31.6.16
Add and .
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Step 4.31.6.16.1
Move .
Step 4.31.6.16.2
To write as a fraction with a common denominator, multiply by .
Step 4.31.6.16.3
Combine the numerators over the common denominator.
Step 4.31.6.17
Add and .
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Step 4.31.6.17.1
Move .
Step 4.31.6.17.2
To write as a fraction with a common denominator, multiply by .
Step 4.31.6.17.3
Combine the numerators over the common denominator.
Step 4.31.6.18
Add and .
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Step 4.31.6.18.1
Move .
Step 4.31.6.18.2
To write as a fraction with a common denominator, multiply by .
Step 4.31.6.18.3
Combine the numerators over the common denominator.
Step 4.31.6.19
To write as a fraction with a common denominator, multiply by .
Step 4.31.6.20
Combine the numerators over the common denominator.
Step 4.31.6.21
Reorder terms.
Step 4.31.6.22
Rewrite in a factored form.
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Step 4.31.6.22.1
Factor out of .
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Step 4.31.6.22.1.1
Factor out of .
Step 4.31.6.22.1.2
Factor out of .
Step 4.31.6.22.1.3
Factor out of .
Step 4.31.6.22.1.4
Factor out of .
Step 4.31.6.22.1.5
Factor out of .
Step 4.31.6.22.1.6
Factor out of .
Step 4.31.6.22.1.7
Factor out of .
Step 4.31.6.22.1.8
Factor out of .
Step 4.31.6.22.1.9
Factor out of .
Step 4.31.6.22.1.10
Factor out of .
Step 4.31.6.22.1.11
Factor out of .
Step 4.31.6.22.2
Combine exponents.
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Step 4.31.6.22.2.1
Multiply by by adding the exponents.
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Step 4.31.6.22.2.1.1
Move .
Step 4.31.6.22.2.1.2
Use the power rule to combine exponents.
Step 4.31.6.22.2.1.3
Combine the numerators over the common denominator.
Step 4.31.6.22.2.1.4
Add and .
Step 4.31.6.22.2.1.5
Divide by .
Step 4.31.6.22.2.2
Simplify .
Step 4.31.6.22.2.3
Multiply by by adding the exponents.
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Step 4.31.6.22.2.3.1
Move .
Step 4.31.6.22.2.3.2
Use the power rule to combine exponents.
Step 4.31.6.22.2.3.3
Combine the numerators over the common denominator.
Step 4.31.6.22.2.3.4
Add and .
Step 4.31.6.22.2.3.5
Divide by .
Step 4.31.6.22.2.4
Simplify .
Step 4.31.6.22.2.5
Multiply by by adding the exponents.
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Step 4.31.6.22.2.5.1
Move .
Step 4.31.6.22.2.5.2
Use the power rule to combine exponents.
Step 4.31.6.22.2.5.3
Combine the numerators over the common denominator.
Step 4.31.6.22.2.5.4
Add and .
Step 4.31.6.22.2.5.5
Divide by .
Step 4.31.6.22.2.6
Simplify .
Step 4.31.6.22.2.7
Multiply by by adding the exponents.
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Step 4.31.6.22.2.7.1
Move .
Step 4.31.6.22.2.7.2
Use the power rule to combine exponents.
Step 4.31.6.22.2.7.3
Combine the numerators over the common denominator.
Step 4.31.6.22.2.7.4
Add and .
Step 4.31.6.22.2.7.5
Divide by .
Step 4.31.6.22.2.8
Simplify .
Step 4.31.6.22.2.9
Multiply by by adding the exponents.
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Step 4.31.6.22.2.9.1
Move .
Step 4.31.6.22.2.9.2
Use the power rule to combine exponents.
Step 4.31.6.22.2.9.3
Combine the numerators over the common denominator.
Step 4.31.6.22.2.9.4
Add and .
Step 4.31.6.22.2.9.5
Divide by .
Step 4.31.6.22.2.10
Simplify .
Step 4.31.6.23
Simplify the numerator.
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Step 4.31.6.23.1
Apply the distributive property.
Step 4.31.6.23.2
Rewrite using the commutative property of multiplication.
Step 4.31.6.23.3
Multiply by .
Step 4.31.6.23.4
Simplify each term.
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Step 4.31.6.23.4.1
Multiply by by adding the exponents.
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Step 4.31.6.23.4.1.1
Move .
Step 4.31.6.23.4.1.2
Multiply by .
Step 4.31.6.23.4.2
Multiply by .
Step 4.31.6.23.5
Apply the distributive property.
Step 4.31.6.23.6
Multiply by .
Step 4.31.6.23.7
Multiply by .
Step 4.31.6.23.8
Apply the distributive property.
Step 4.31.6.23.9
Rewrite using the commutative property of multiplication.
Step 4.31.6.23.10
Multiply by .
Step 4.31.6.23.11
Simplify each term.
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Step 4.31.6.23.11.1
Multiply by by adding the exponents.
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Step 4.31.6.23.11.1.1
Move .
Step 4.31.6.23.11.1.2
Multiply by .
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Step 4.31.6.23.11.1.2.1
Raise to the power of .
Step 4.31.6.23.11.1.2.2
Use the power rule to combine exponents.
Step 4.31.6.23.11.1.3
Add and .
Step 4.31.6.23.11.2
Multiply by .
Step 4.31.6.23.12
Apply the distributive property.
Step 4.31.6.23.13
Rewrite using the commutative property of multiplication.
Step 4.31.6.23.14
Multiply by .
Step 4.31.6.23.15
Simplify each term.
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Step 4.31.6.23.15.1
Multiply by by adding the exponents.
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Step 4.31.6.23.15.1.1
Move .
Step 4.31.6.23.15.1.2
Multiply by .
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Step 4.31.6.23.15.1.2.1
Raise to the power of .
Step 4.31.6.23.15.1.2.2
Use the power rule to combine exponents.
Step 4.31.6.23.15.1.3
Add and .
Step 4.31.6.23.15.2
Multiply by .
Step 4.31.6.23.16
Apply the distributive property.
Step 4.31.6.23.17
Rewrite using the commutative property of multiplication.
Step 4.31.6.23.18
Multiply by .
Step 4.31.6.23.19
Simplify each term.
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Step 4.31.6.23.19.1
Multiply by by adding the exponents.
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Step 4.31.6.23.19.1.1
Move .
Step 4.31.6.23.19.1.2
Multiply by .
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Step 4.31.6.23.19.1.2.1
Raise to the power of .
Step 4.31.6.23.19.1.2.2
Use the power rule to combine exponents.
Step 4.31.6.23.19.1.3
Add and .
Step 4.31.6.23.19.2
Multiply by .
Step 4.31.6.23.20
Rewrite as .
Step 4.31.6.23.21
Expand using the FOIL Method.
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Step 4.31.6.23.21.1
Apply the distributive property.
Step 4.31.6.23.21.2
Apply the distributive property.
Step 4.31.6.23.21.3
Apply the distributive property.
Step 4.31.6.23.22
Simplify and combine like terms.
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Step 4.31.6.23.22.1
Simplify each term.
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Step 4.31.6.23.22.1.1
Rewrite using the commutative property of multiplication.
Step 4.31.6.23.22.1.2
Multiply by by adding the exponents.
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Step 4.31.6.23.22.1.2.1
Move .
Step 4.31.6.23.22.1.2.2
Multiply by .
Step 4.31.6.23.22.1.3
Multiply by .
Step 4.31.6.23.22.1.4
Multiply by .
Step 4.31.6.23.22.1.5
Multiply by .
Step 4.31.6.23.22.1.6
Multiply by .
Step 4.31.6.23.22.2
Add and .
Step 4.31.6.23.23
Expand by multiplying each term in the first expression by each term in the second expression.
Step 4.31.6.23.24
Simplify each term.
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Step 4.31.6.23.24.1
Rewrite using the commutative property of multiplication.
Step 4.31.6.23.24.2
Multiply by by adding the exponents.
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Step 4.31.6.23.24.2.1
Move .
Step 4.31.6.23.24.2.2
Multiply by .
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Step 4.31.6.23.24.2.2.1
Raise to the power of .
Step 4.31.6.23.24.2.2.2
Use the power rule to combine exponents.
Step 4.31.6.23.24.2.3
Add and .
Step 4.31.6.23.24.3
Multiply by .
Step 4.31.6.23.24.4
Multiply by .
Step 4.31.6.23.24.5
Rewrite using the commutative property of multiplication.
Step 4.31.6.23.24.6
Multiply by by adding the exponents.
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Step 4.31.6.23.24.6.1
Move .
Step 4.31.6.23.24.6.2
Multiply by .
Step 4.31.6.23.24.7
Multiply by .
Step 4.31.6.23.24.8
Multiply by .
Step 4.31.6.23.24.9
Multiply by .
Step 4.31.6.23.24.10
Multiply by .
Step 4.31.6.23.25
Subtract from .
Step 4.31.6.23.26
Subtract from .
Step 4.31.6.23.27
Add and .
Step 4.31.6.23.28
Add and .
Step 4.31.6.23.29
Subtract from .
Step 4.31.6.23.30
Subtract from .
Step 4.31.6.23.31
Add and .
Step 4.31.6.23.32
Add and .
Step 4.31.6.23.33
Subtract from .
Step 4.31.6.23.34
Subtract from .
Step 4.31.6.23.35
Reorder terms.
Step 4.31.7
Combine terms.
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Step 4.31.7.1
Multiply by .
Step 4.31.7.2
Rewrite as a product.
Step 4.31.7.3
Multiply by .
Step 4.31.7.4
Multiply by by adding the exponents.
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Step 4.31.7.4.1
Move .
Step 4.31.7.4.2
Use the power rule to combine exponents.
Step 4.31.7.4.3
Combine the numerators over the common denominator.
Step 4.31.7.4.4
Add and .
Step 4.31.7.4.5
Divide by .
Step 4.31.7.5
Simplify .
Step 4.31.8
Reorder terms.
Step 4.31.9
Factor out of .
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Step 4.31.9.1
Multiply by .
Step 4.31.9.2
Factor out of .
Step 4.31.10
Cancel the common factor of .
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Step 4.31.10.1
Cancel the common factor.
Step 4.31.10.2
Rewrite the expression.
Step 5
Reform the equation by setting the left side equal to the right side.
Step 6
Replace with .