Calculus Examples

Find the Derivative - d/dx y=(x^2+1)^( natural log of x)
Step 1
Use the properties of logarithms to simplify the differentiation.
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Step 1.1
Rewrite as .
Step 1.2
Expand by moving outside the logarithm.
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
Differentiate using the Exponential Rule which states that is where =.
Step 2.3
Replace all occurrences of with .
Step 3
Differentiate using the Product Rule which states that is where and .
Step 4
Differentiate using the chain rule, which states that is where and .
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Step 4.1
To apply the Chain Rule, set as .
Step 4.2
The derivative of with respect to is .
Step 4.3
Replace all occurrences of with .
Step 5
Differentiate.
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Step 5.1
Combine and .
Step 5.2
By the Sum Rule, the derivative of with respect to is .
Step 5.3
Differentiate using the Power Rule which states that is where .
Step 5.4
Since is constant with respect to , the derivative of with respect to is .
Step 5.5
Combine fractions.
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Step 5.5.1
Add and .
Step 5.5.2
Combine and .
Step 5.5.3
Combine and .
Step 6
The derivative of with respect to is .
Step 7
Combine and .
Step 8
To write as a fraction with a common denominator, multiply by .
Step 9
To write as a fraction with a common denominator, multiply by .
Step 10
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 10.1
Multiply by .
Step 10.2
Multiply by .
Step 10.3
Reorder the factors of .
Step 11
Combine the numerators over the common denominator.
Step 12
Raise to the power of .
Step 13
Raise to the power of .
Step 14
Use the power rule to combine exponents.
Step 15
Add and .
Step 16
Combine and .
Step 17
Simplify.
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Step 17.1
Apply the distributive property.
Step 17.2
Simplify the numerator.
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Step 17.2.1
Simplify each term.
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Step 17.2.1.1
Simplify by moving inside the logarithm.
Step 17.2.1.2
Apply the distributive property.
Step 17.2.1.3
Multiply by .
Step 17.2.2
Apply the distributive property.
Step 17.2.3
Reorder factors in .
Step 17.3
Combine terms.
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Step 17.3.1
Multiply by by adding the exponents.
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Step 17.3.1.1
Multiply by .
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Step 17.3.1.1.1
Raise to the power of .
Step 17.3.1.1.2
Use the power rule to combine exponents.
Step 17.3.1.2
Add and .
Step 17.3.2
Multiply by .
Step 17.4
Reorder terms.