Calculus Examples

Find the Derivative - d/dx y=(x^3-2x)^( 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
Differentiate using the Power Rule which states that is where .
Step 5.6
Multiply by .
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
Combine and .
Step 10
Combine the numerators over the common denominator.
Step 11
Combine and .
Step 12
Combine and .
Step 13
Simplify.
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Step 13.1
Simplify the numerator.
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Step 13.1.1
Simplify each term.
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Step 13.1.1.1
Factor out of .
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Step 13.1.1.1.1
Factor out of .
Step 13.1.1.1.2
Factor out of .
Step 13.1.1.1.3
Factor out of .
Step 13.1.1.2
Cancel the common factor of .
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Step 13.1.1.2.1
Cancel the common factor.
Step 13.1.1.2.2
Rewrite the expression.
Step 13.1.1.3
Multiply by .
Step 13.1.2
To write as a fraction with a common denominator, multiply by .
Step 13.1.3
Combine the numerators over the common denominator.
Step 13.1.4
Simplify the numerator.
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Step 13.1.4.1
Apply the distributive property.
Step 13.1.4.2
Rewrite using the commutative property of multiplication.
Step 13.1.4.3
Multiply .
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Step 13.1.4.3.1
Reorder and .
Step 13.1.4.3.2
Simplify by moving inside the logarithm.
Step 13.1.4.4
Simplify by moving inside the logarithm.
Step 13.1.4.5
Apply the distributive property.
Step 13.1.4.6
Multiply .
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Step 13.1.4.6.1
Reorder and .
Step 13.1.4.6.2
Simplify by moving inside the logarithm.
Step 13.1.5
Combine and .
Step 13.1.6
Reorder factors in .
Step 13.2
Combine terms.
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Step 13.2.1
Rewrite as a product.
Step 13.2.2
Multiply by .
Step 13.3
Reorder terms.