Calculus Examples

Find the Derivative - d/dx ( natural log of x)/( natural log of x+1)
Step 1
Differentiate using the Quotient Rule which states that is where and .
Step 2
The derivative of with respect to is .
Step 3
Combine and .
Step 4
Multiply by .
Step 5
Simplify terms.
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Step 5.1
Combine.
Step 5.2
Apply the distributive property.
Step 5.3
Cancel the common factor of .
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Step 5.3.1
Cancel the common factor.
Step 5.3.2
Rewrite the expression.
Step 6
Differentiate using the chain rule, which states that is where and .
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Step 6.1
To apply the Chain Rule, set as .
Step 6.2
The derivative of with respect to is .
Step 6.3
Replace all occurrences of with .
Step 7
Differentiate.
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Step 7.1
Combine and .
Step 7.2
Combine and .
Step 7.3
By the Sum Rule, the derivative of with respect to is .
Step 7.4
Differentiate using the Power Rule which states that is where .
Step 7.5
Since is constant with respect to , the derivative of with respect to is .
Step 7.6
Simplify the expression.
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Step 7.6.1
Add and .
Step 7.6.2
Multiply by .
Step 8
To write as a fraction with a common denominator, multiply by .
Step 9
Combine the numerators over the common denominator.
Step 10
Rewrite as a product.
Step 11
Multiply by .
Step 12
Simplify.
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Step 12.1
Apply the distributive property.
Step 12.2
Simplify the numerator.
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Step 12.2.1
Simplify each term.
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Step 12.2.1.1
Apply the distributive property.
Step 12.2.1.2
Multiply by .
Step 12.2.2
Reorder factors in .
Step 12.3
Combine terms.
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Step 12.3.1
Raise to the power of .
Step 12.3.2
Raise to the power of .
Step 12.3.3
Use the power rule to combine exponents.
Step 12.3.4
Add and .
Step 12.3.5
Multiply by .
Step 12.4
Reorder terms.