Calculus Examples

Find the Derivative - d/d@VAR p(x) = natural log of -x^3+3x^2+72x+1
Step 1
Differentiate using the chain rule, which states that is where and .
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Step 1.1
To apply the Chain Rule, set as .
Step 1.2
The derivative of with respect to is .
Step 1.3
Replace all occurrences of with .
Step 2
Differentiate.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.3
Differentiate using the Power Rule which states that is where .
Step 2.4
Multiply by .
Step 2.5
Since is constant with respect to , the derivative of with respect to is .
Step 2.6
Differentiate using the Power Rule which states that is where .
Step 2.7
Multiply by .
Step 2.8
Since is constant with respect to , the derivative of with respect to is .
Step 2.9
Differentiate using the Power Rule which states that is where .
Step 2.10
Multiply by .
Step 2.11
Since is constant with respect to , the derivative of with respect to is .
Step 2.12
Add and .
Step 3
Simplify.
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Step 3.1
Reorder the factors of .
Step 3.2
Multiply by .
Step 3.3
Simplify the numerator.
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Step 3.3.1
Factor out of .
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Step 3.3.1.1
Factor out of .
Step 3.3.1.2
Factor out of .
Step 3.3.1.3
Factor out of .
Step 3.3.1.4
Factor out of .
Step 3.3.1.5
Factor out of .
Step 3.3.2
Factor by grouping.
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Step 3.3.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 3.3.2.1.1
Factor out of .
Step 3.3.2.1.2
Rewrite as plus
Step 3.3.2.1.3
Apply the distributive property.
Step 3.3.2.2
Factor out the greatest common factor from each group.
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Step 3.3.2.2.1
Group the first two terms and the last two terms.
Step 3.3.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 3.3.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 3.4
Factor out of .
Step 3.5
Rewrite as .
Step 3.6
Factor out of .
Step 3.7
Rewrite as .
Step 3.8
Factor out of .
Step 3.9
Factor out of .
Step 3.10
Factor out of .
Step 3.11
Factor out of .
Step 3.12
Factor out of .
Step 3.13
Rewrite as .
Step 3.14
Factor out of .
Step 3.15
Rewrite as .
Step 3.16
Cancel the common factor.
Step 3.17
Rewrite the expression.