Calculus Examples

Find the Derivative - d/dx f(x)=x natural log of 1+x^2
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
Combine and .
Step 3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4
Add and .
Step 3.5
Differentiate using the Power Rule which states that is where .
Step 3.6
Combine fractions.
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Step 3.6.1
Combine and .
Step 3.6.2
Combine and .
Step 4
Raise to the power of .
Step 5
Raise to the power of .
Step 6
Use the power rule to combine exponents.
Step 7
Add and .
Step 8
Differentiate using the Power Rule which states that is where .
Step 9
Multiply by .
Step 10
To write as a fraction with a common denominator, multiply by .
Step 11
Combine the numerators over the common denominator.
Step 12
Simplify.
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Step 12.1
Simplify the numerator.
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Step 12.1.1
Simplify each term.
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Step 12.1.1.1
Apply the distributive property.
Step 12.1.1.2
Multiply by .
Step 12.1.2
Reorder factors in .
Step 12.2
Reorder terms.