Calculus Examples

Find the 2nd Derivative x^2e^(-x)
Step 1
Find the first derivative.
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Step 1.1
Differentiate using the Product Rule which states that is where and .
Step 1.2
Differentiate using the chain rule, which states that is where and .
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Step 1.2.1
To apply the Chain Rule, set as .
Step 1.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.2.3
Replace all occurrences of with .
Step 1.3
Differentiate.
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Step 1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.3.3
Simplify the expression.
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Step 1.3.3.1
Multiply by .
Step 1.3.3.2
Move to the left of .
Step 1.3.3.3
Rewrite as .
Step 1.3.4
Differentiate using the Power Rule which states that is where .
Step 1.4
Simplify.
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Step 1.4.1
Reorder terms.
Step 1.4.2
Reorder factors in .
Step 2
Find the second derivative.
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Step 2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2
Evaluate .
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Step 2.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.2
Differentiate using the Product Rule which states that is where and .
Step 2.2.3
Differentiate using the chain rule, which states that is where and .
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Step 2.2.3.1
To apply the Chain Rule, set as .
Step 2.2.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.2.3.3
Replace all occurrences of with .
Step 2.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.5
Differentiate using the Power Rule which states that is where .
Step 2.2.6
Differentiate using the Power Rule which states that is where .
Step 2.2.7
Multiply by .
Step 2.2.8
Move to the left of .
Step 2.2.9
Rewrite as .
Step 2.3
Evaluate .
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Step 2.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.2
Differentiate using the Product Rule which states that is where and .
Step 2.3.3
Differentiate using the chain rule, which states that is where and .
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Step 2.3.3.1
To apply the Chain Rule, set as .
Step 2.3.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 2.3.3.3
Replace all occurrences of with .
Step 2.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 2.3.5
Differentiate using the Power Rule which states that is where .
Step 2.3.6
Differentiate using the Power Rule which states that is where .
Step 2.3.7
Multiply by .
Step 2.3.8
Move to the left of .
Step 2.3.9
Rewrite as .
Step 2.3.10
Multiply by .
Step 2.4
Simplify.
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Step 2.4.1
Apply the distributive property.
Step 2.4.2
Apply the distributive property.
Step 2.4.3
Combine terms.
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Step 2.4.3.1
Multiply by .
Step 2.4.3.2
Multiply by .
Step 2.4.3.3
Multiply by .
Step 2.4.3.4
Multiply by .
Step 2.4.3.5
Subtract from .
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Step 2.4.3.5.1
Move .
Step 2.4.3.5.2
Subtract from .
Step 2.4.4
Reorder terms.
Step 2.4.5
Reorder factors in .
Step 3
Find the third derivative.
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Step 3.1
By the Sum Rule, the derivative of with respect to is .
Step 3.2
Evaluate .
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Step 3.2.1
Differentiate using the Product Rule which states that is where and .
Step 3.2.2
Differentiate using the chain rule, which states that is where and .
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Step 3.2.2.1
To apply the Chain Rule, set as .
Step 3.2.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.2.2.3
Replace all occurrences of with .
Step 3.2.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.2.4
Differentiate using the Power Rule which states that is where .
Step 3.2.5
Differentiate using the Power Rule which states that is where .
Step 3.2.6
Multiply by .
Step 3.2.7
Move to the left of .
Step 3.2.8
Rewrite as .
Step 3.3
Evaluate .
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Step 3.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.2
Differentiate using the Product Rule which states that is where and .
Step 3.3.3
Differentiate using the chain rule, which states that is where and .
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Step 3.3.3.1
To apply the Chain Rule, set as .
Step 3.3.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.3.3.3
Replace all occurrences of with .
Step 3.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.5
Differentiate using the Power Rule which states that is where .
Step 3.3.6
Differentiate using the Power Rule which states that is where .
Step 3.3.7
Multiply by .
Step 3.3.8
Move to the left of .
Step 3.3.9
Rewrite as .
Step 3.3.10
Multiply by .
Step 3.4
Evaluate .
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Step 3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.2
Differentiate using the chain rule, which states that is where and .
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Step 3.4.2.1
To apply the Chain Rule, set as .
Step 3.4.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.4.2.3
Replace all occurrences of with .
Step 3.4.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.4.4
Differentiate using the Power Rule which states that is where .
Step 3.4.5
Multiply by .
Step 3.4.6
Move to the left of .
Step 3.4.7
Rewrite as .
Step 3.4.8
Multiply by .
Step 3.5
Simplify.
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Step 3.5.1
Apply the distributive property.
Step 3.5.2
Combine terms.
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Step 3.5.2.1
Multiply by .
Step 3.5.2.2
Add and .
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Step 3.5.2.2.1
Move .
Step 3.5.2.2.2
Add and .
Step 3.5.2.3
Subtract from .
Step 3.5.3
Reorder terms.
Step 3.5.4
Reorder factors in .
Step 4
Find the fourth derivative.
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Step 4.1
By the Sum Rule, the derivative of with respect to is .
Step 4.2
Evaluate .
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Step 4.2.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2.2
Differentiate using the Product Rule which states that is where and .
Step 4.2.3
Differentiate using the chain rule, which states that is where and .
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Step 4.2.3.1
To apply the Chain Rule, set as .
Step 4.2.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.2.3.3
Replace all occurrences of with .
Step 4.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.2.5
Differentiate using the Power Rule which states that is where .
Step 4.2.6
Differentiate using the Power Rule which states that is where .
Step 4.2.7
Multiply by .
Step 4.2.8
Move to the left of .
Step 4.2.9
Rewrite as .
Step 4.3
Evaluate .
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Step 4.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.2
Differentiate using the Product Rule which states that is where and .
Step 4.3.3
Differentiate using the chain rule, which states that is where and .
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Step 4.3.3.1
To apply the Chain Rule, set as .
Step 4.3.3.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.3.3.3
Replace all occurrences of with .
Step 4.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 4.3.5
Differentiate using the Power Rule which states that is where .
Step 4.3.6
Differentiate using the Power Rule which states that is where .
Step 4.3.7
Multiply by .
Step 4.3.8
Move to the left of .
Step 4.3.9
Rewrite as .
Step 4.3.10
Multiply by .
Step 4.4
Evaluate .
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Step 4.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.4.2
Differentiate using the chain rule, which states that is where and .
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Step 4.4.2.1
To apply the Chain Rule, set as .
Step 4.4.2.2
Differentiate using the Exponential Rule which states that is where =.
Step 4.4.2.3
Replace all occurrences of with .
Step 4.4.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.4.4
Differentiate using the Power Rule which states that is where .
Step 4.4.5
Multiply by .
Step 4.4.6
Move to the left of .
Step 4.4.7
Rewrite as .
Step 4.4.8
Multiply by .
Step 4.5
Simplify.
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Step 4.5.1
Apply the distributive property.
Step 4.5.2
Apply the distributive property.
Step 4.5.3
Combine terms.
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Step 4.5.3.1
Multiply by .
Step 4.5.3.2
Multiply by .
Step 4.5.3.3
Multiply by .
Step 4.5.3.4
Multiply by .
Step 4.5.3.5
Subtract from .
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Step 4.5.3.5.1
Move .
Step 4.5.3.5.2
Subtract from .
Step 4.5.3.6
Add and .
Step 4.5.4
Reorder terms.
Step 4.5.5
Reorder factors in .