Calculus Examples

Find the Derivative - d/dx y=(x^3+1) natural log of x^3+1
Step 1
Differentiate using the Product Rule which states that is where and .
Step 2
Differentiate using the chain rule, which states that is where and .
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Step 2.1
To apply the Chain Rule, set as .
Step 2.2
The derivative of with respect to is .
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Differentiate using the Power Rule which states that is where .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Combine fractions.
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Step 3.4.1
Add and .
Step 3.4.2
Combine and .
Step 3.4.3
Combine and .
Step 3.4.4
Move to the left of .
Step 3.5
By the Sum Rule, the derivative of with respect to is .
Step 3.6
Differentiate using the Power Rule which states that is where .
Step 3.7
Since is constant with respect to , the derivative of with respect to is .
Step 3.8
Add and .
Step 4
Simplify.
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Step 4.1
Reorder terms.
Step 4.2
Simplify each term.
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Step 4.2.1
Simplify the denominator.
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Step 4.2.1.1
Rewrite as .
Step 4.2.1.2
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Step 4.2.1.3
Simplify.
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Step 4.2.1.3.1
Multiply by .
Step 4.2.1.3.2
One to any power is one.
Step 4.2.2
Multiply by .
Step 4.2.3
Simplify the numerator.
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Step 4.2.3.1
Rewrite as .
Step 4.2.3.2
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Step 4.2.3.3
Simplify.
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Step 4.2.3.3.1
Multiply by .
Step 4.2.3.3.2
One to any power is one.
Step 4.2.4
Cancel the common factor of .
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Step 4.2.4.1
Cancel the common factor.
Step 4.2.4.2
Rewrite the expression.
Step 4.2.5
Cancel the common factor of .
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Step 4.2.5.1
Cancel the common factor.
Step 4.2.5.2
Divide by .