Calculus Examples

Find the Derivative - d/d@VAR f(x)=(4x+3)^2 natural log of 4x+3
Step 1
Rewrite as .
Step 2
Expand using the FOIL Method.
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Step 2.1
Apply the distributive property.
Step 2.2
Apply the distributive property.
Step 2.3
Apply the distributive property.
Step 3
Simplify and combine like terms.
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Step 3.1
Simplify each term.
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Step 3.1.1
Rewrite using the commutative property of multiplication.
Step 3.1.2
Multiply by by adding the exponents.
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Step 3.1.2.1
Move .
Step 3.1.2.2
Multiply by .
Step 3.1.3
Multiply by .
Step 3.1.4
Multiply by .
Step 3.1.5
Multiply by .
Step 3.1.6
Multiply by .
Step 3.2
Add and .
Step 4
Differentiate using the Product Rule which states that is where and .
Step 5
Differentiate using the chain rule, which states that is where and .
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Step 5.1
To apply the Chain Rule, set as .
Step 5.2
The derivative of with respect to is .
Step 5.3
Replace all occurrences of with .
Step 6
Differentiate.
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Step 6.1
By the Sum Rule, the derivative of with respect to is .
Step 6.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.3
Differentiate using the Power Rule which states that is where .
Step 6.4
Multiply by .
Step 6.5
Since is constant with respect to , the derivative of with respect to is .
Step 6.6
Combine fractions.
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Step 6.6.1
Add and .
Step 6.6.2
Combine and .
Step 6.7
By the Sum Rule, the derivative of with respect to is .
Step 6.8
Since is constant with respect to , the derivative of with respect to is .
Step 6.9
Differentiate using the Power Rule which states that is where .
Step 6.10
Multiply by .
Step 6.11
Since is constant with respect to , the derivative of with respect to is .
Step 6.12
Differentiate using the Power Rule which states that is where .
Step 6.13
Multiply by .
Step 6.14
Since is constant with respect to , the derivative of with respect to is .
Step 6.15
Add and .
Step 7
Simplify.
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Step 7.1
Reorder terms.
Step 7.2
Simplify each term.
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Step 7.2.1
Multiply by .
Step 7.2.2
Factor using the perfect square rule.
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Step 7.2.2.1
Rewrite as .
Step 7.2.2.2
Rewrite as .
Step 7.2.2.3
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 7.2.2.4
Rewrite the polynomial.
Step 7.2.2.5
Factor using the perfect square trinomial rule , where and .
Step 7.2.3
Cancel the common factor of and .
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Step 7.2.3.1
Factor out of .
Step 7.2.3.2
Cancel the common factors.
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Step 7.2.3.2.1
Multiply by .
Step 7.2.3.2.2
Cancel the common factor.
Step 7.2.3.2.3
Rewrite the expression.
Step 7.2.3.2.4
Divide by .
Step 7.2.4
Apply the distributive property.
Step 7.2.5
Multiply by .
Step 7.2.6
Multiply by .
Step 7.2.7
Apply the distributive property.