Calculus Examples

Find the 2nd Derivative f(x)=24/(x^2+12)
Step 1
Find the first derivative.
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Step 1.1
Differentiate using the Constant Multiple Rule.
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Step 1.1.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.2
Rewrite as .
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 Power 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
Multiply by .
Step 1.3.2
By the Sum Rule, the derivative of with respect to is .
Step 1.3.3
Differentiate using the Power Rule which states that is where .
Step 1.3.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.5
Simplify the expression.
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Step 1.3.5.1
Add and .
Step 1.3.5.2
Multiply by .
Step 1.4
Simplify.
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Step 1.4.1
Rewrite the expression using the negative exponent rule .
Step 1.4.2
Combine terms.
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Step 1.4.2.1
Combine and .
Step 1.4.2.2
Move the negative in front of the fraction.
Step 1.4.2.3
Combine and .
Step 1.4.2.4
Move to the left of .
Step 2
Find the second derivative.
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Step 2.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2
Differentiate using the Quotient Rule which states that is where and .
Step 2.3
Differentiate using the Power Rule.
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Step 2.3.1
Multiply the exponents in .
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Step 2.3.1.1
Apply the power rule and multiply exponents, .
Step 2.3.1.2
Multiply by .
Step 2.3.2
Differentiate using the Power Rule which states that is where .
Step 2.3.3
Multiply by .
Step 2.4
Differentiate using the chain rule, which states that is where and .
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Step 2.4.1
To apply the Chain Rule, set as .
Step 2.4.2
Differentiate using the Power Rule which states that is where .
Step 2.4.3
Replace all occurrences of with .
Step 2.5
Simplify with factoring out.
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Step 2.5.1
Multiply by .
Step 2.5.2
Factor out of .
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Step 2.5.2.1
Factor out of .
Step 2.5.2.2
Factor out of .
Step 2.5.2.3
Factor out of .
Step 2.6
Cancel the common factors.
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Step 2.6.1
Factor out of .
Step 2.6.2
Cancel the common factor.
Step 2.6.3
Rewrite the expression.
Step 2.7
By the Sum Rule, the derivative of with respect to is .
Step 2.8
Differentiate using the Power Rule which states that is where .
Step 2.9
Since is constant with respect to , the derivative of with respect to is .
Step 2.10
Simplify the expression.
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Step 2.10.1
Add and .
Step 2.10.2
Multiply by .
Step 2.11
Raise to the power of .
Step 2.12
Raise to the power of .
Step 2.13
Use the power rule to combine exponents.
Step 2.14
Add and .
Step 2.15
Subtract from .
Step 2.16
Combine and .
Step 2.17
Move the negative in front of the fraction.
Step 2.18
Simplify.
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Step 2.18.1
Apply the distributive property.
Step 2.18.2
Simplify each term.
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Step 2.18.2.1
Multiply by .
Step 2.18.2.2
Multiply by .
Step 3
Find the third derivative.
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Step 3.1
Differentiate using the Product Rule which states that is where and .
Step 3.2
Differentiate using the Quotient Rule which states that is where and .
Step 3.3
Differentiate.
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Step 3.3.1
Multiply the exponents in .
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Step 3.3.1.1
Apply the power rule and multiply exponents, .
Step 3.3.1.2
Multiply by .
Step 3.3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.4
Differentiate using the Power Rule which states that is where .
Step 3.3.5
Multiply by .
Step 3.3.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.3.7
Add and .
Step 3.4
Differentiate using the chain rule, which states that is where and .
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Step 3.4.1
To apply the Chain Rule, set as .
Step 3.4.2
Differentiate using the Power Rule which states that is where .
Step 3.4.3
Replace all occurrences of with .
Step 3.5
Differentiate.
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Step 3.5.1
Multiply by .
Step 3.5.2
By the Sum Rule, the derivative of with respect to is .
Step 3.5.3
Differentiate using the Power Rule which states that is where .
Step 3.5.4
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.5
Simplify the expression.
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Step 3.5.5.1
Add and .
Step 3.5.5.2
Move to the left of .
Step 3.5.5.3
Multiply by .
Step 3.5.6
Since is constant with respect to , the derivative of with respect to is .
Step 3.5.7
Simplify the expression.
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Step 3.5.7.1
Multiply by .
Step 3.5.7.2
Add and .
Step 3.6
Simplify.
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Step 3.6.1
Apply the distributive property.
Step 3.6.2
Simplify the numerator.
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Step 3.6.2.1
Factor out of .
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Step 3.6.2.1.1
Factor out of .
Step 3.6.2.1.2
Factor out of .
Step 3.6.2.1.3
Factor out of .
Step 3.6.2.2
Combine exponents.
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Step 3.6.2.2.1
Multiply by .
Step 3.6.2.2.2
Multiply by .
Step 3.6.2.3
Simplify each term.
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Step 3.6.2.3.1
Apply the distributive property.
Step 3.6.2.3.2
Move to the left of .
Step 3.6.2.3.3
Multiply by .
Step 3.6.2.4
Add and .
Step 3.6.2.5
Subtract from .
Step 3.6.2.6
Factor out of .
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Step 3.6.2.6.1
Factor out of .
Step 3.6.2.6.2
Factor out of .
Step 3.6.2.6.3
Factor out of .
Step 3.6.3
Combine terms.
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Step 3.6.3.1
Move to the left of .
Step 3.6.3.2
Cancel the common factor of and .
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Step 3.6.3.2.1
Factor out of .
Step 3.6.3.2.2
Cancel the common factors.
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Step 3.6.3.2.2.1
Factor out of .
Step 3.6.3.2.2.2
Cancel the common factor.
Step 3.6.3.2.2.3
Rewrite the expression.
Step 4
Find the fourth derivative.
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Step 4.1
Since is constant with respect to , the derivative of with respect to is .
Step 4.2
Differentiate using the Quotient Rule which states that is where and .
Step 4.3
Multiply the exponents in .
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Step 4.3.1
Apply the power rule and multiply exponents, .
Step 4.3.2
Multiply by .
Step 4.4
Differentiate using the Product Rule which states that is where and .
Step 4.5
Differentiate.
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Step 4.5.1
By the Sum Rule, the derivative of with respect to is .
Step 4.5.2
Differentiate using the Power Rule which states that is where .
Step 4.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.5.4
Add and .
Step 4.6
Raise to the power of .
Step 4.7
Raise to the power of .
Step 4.8
Use the power rule to combine exponents.
Step 4.9
Differentiate using the Power Rule.
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Step 4.9.1
Add and .
Step 4.9.2
Differentiate using the Power Rule which states that is where .
Step 4.9.3
Simplify by adding terms.
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Step 4.9.3.1
Multiply by .
Step 4.9.3.2
Add and .
Step 4.10
Differentiate using the chain rule, which states that is where and .
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Step 4.10.1
To apply the Chain Rule, set as .
Step 4.10.2
Differentiate using the Power Rule which states that is where .
Step 4.10.3
Replace all occurrences of with .
Step 4.11
Simplify with factoring out.
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Step 4.11.1
Multiply by .
Step 4.11.2
Factor out of .
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Step 4.11.2.1
Factor out of .
Step 4.11.2.2
Factor out of .
Step 4.11.2.3
Factor out of .
Step 4.12
Cancel the common factors.
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Step 4.12.1
Factor out of .
Step 4.12.2
Cancel the common factor.
Step 4.12.3
Rewrite the expression.
Step 4.13
By the Sum Rule, the derivative of with respect to is .
Step 4.14
Differentiate using the Power Rule which states that is where .
Step 4.15
Since is constant with respect to , the derivative of with respect to is .
Step 4.16
Simplify the expression.
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Step 4.16.1
Add and .
Step 4.16.2
Multiply by .
Step 4.17
Raise to the power of .
Step 4.18
Raise to the power of .
Step 4.19
Use the power rule to combine exponents.
Step 4.20
Add and .
Step 4.21
Combine and .
Step 4.22
Move the negative in front of the fraction.
Step 4.23
Simplify.
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Step 4.23.1
Apply the distributive property.
Step 4.23.2
Apply the distributive property.
Step 4.23.3
Simplify the numerator.
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Step 4.23.3.1
Simplify each term.
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Step 4.23.3.1.1
Expand using the FOIL Method.
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Step 4.23.3.1.1.1
Apply the distributive property.
Step 4.23.3.1.1.2
Apply the distributive property.
Step 4.23.3.1.1.3
Apply the distributive property.
Step 4.23.3.1.2
Simplify and combine like terms.
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Step 4.23.3.1.2.1
Simplify each term.
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Step 4.23.3.1.2.1.1
Rewrite using the commutative property of multiplication.
Step 4.23.3.1.2.1.2
Multiply by by adding the exponents.
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Step 4.23.3.1.2.1.2.1
Move .
Step 4.23.3.1.2.1.2.2
Use the power rule to combine exponents.
Step 4.23.3.1.2.1.2.3
Add and .
Step 4.23.3.1.2.1.3
Move to the left of .
Step 4.23.3.1.2.1.4
Multiply by .
Step 4.23.3.1.2.1.5
Multiply by .
Step 4.23.3.1.2.2
Add and .
Step 4.23.3.1.3
Apply the distributive property.
Step 4.23.3.1.4
Simplify.
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Step 4.23.3.1.4.1
Multiply by .
Step 4.23.3.1.4.2
Multiply by .
Step 4.23.3.1.4.3
Multiply by .
Step 4.23.3.1.5
Multiply by by adding the exponents.
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Step 4.23.3.1.5.1
Move .
Step 4.23.3.1.5.2
Use the power rule to combine exponents.
Step 4.23.3.1.5.3
Add and .
Step 4.23.3.1.6
Multiply by .
Step 4.23.3.1.7
Multiply by .
Step 4.23.3.1.8
Multiply by .
Step 4.23.3.2
Subtract from .
Step 4.23.3.3
Add and .
Step 4.23.4
Factor out of .
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Step 4.23.4.1
Factor out of .
Step 4.23.4.2
Factor out of .
Step 4.23.4.3
Factor out of .
Step 4.23.4.4
Factor out of .
Step 4.23.4.5
Factor out of .
Step 4.23.5
Factor out of .
Step 4.23.6
Factor out of .
Step 4.23.7
Factor out of .
Step 4.23.8
Rewrite as .
Step 4.23.9
Factor out of .
Step 4.23.10
Rewrite as .
Step 4.23.11
Move the negative in front of the fraction.
Step 4.23.12
Multiply by .
Step 4.23.13
Multiply by .
Step 5
The fourth derivative of with respect to is .