Calculus Examples

Find the Derivative - d/dx y=x^2+x^4e^(-2 natural log of x)
Step 1
Differentiate.
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Step 1.1
By the Sum Rule, the derivative of with respect to is .
Step 1.2
Differentiate using the Power Rule which states that is where .
Step 2
Evaluate .
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Step 2.1
Differentiate using the Product Rule which states that is where and .
Step 2.2
Differentiate using the chain rule, which states that is where and .
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Step 2.2.1
To apply the Chain Rule, set as .
Step 2.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.2.3
Replace all occurrences of with .
Step 2.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.4
The derivative of with respect to is .
Step 2.5
Differentiate using the Power Rule which states that is where .
Step 2.6
Combine and .
Step 2.7
Move the negative in front of the fraction.
Step 2.8
Combine and .
Step 2.9
Move to the left of .
Step 2.10
Combine and .
Step 2.11
Cancel the common factor of and .
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Step 2.11.1
Factor out of .
Step 2.11.2
Cancel the common factors.
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Step 2.11.2.1
Raise to the power of .
Step 2.11.2.2
Factor out of .
Step 2.11.2.3
Cancel the common factor.
Step 2.11.2.4
Rewrite the expression.
Step 2.11.2.5
Divide by .
Step 2.12
Multiply by .
Step 2.13
Add and .
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Step 2.13.1
Move .
Step 2.13.2
Add and .
Step 3
Simplify.
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Step 3.1
Reorder terms.
Step 3.2
Reorder factors in .