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Calculus Examples
Step 1
Step 1.1
Differentiate using the Quotient Rule which states that is where and .
Step 1.2
Differentiate.
Step 1.2.1
Differentiate using the Power Rule which states that is where .
Step 1.2.2
Move to the left of .
Step 1.2.3
By the Sum Rule, the derivative of with respect to is .
Step 1.2.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.5
Add and .
Step 1.2.6
Since is constant with respect to , the derivative of with respect to is .
Step 1.2.7
Multiply by .
Step 1.2.8
Differentiate using the Power Rule which states that is where .
Step 1.2.9
Multiply by .
Step 1.3
Simplify.
Step 1.3.1
Apply the distributive property.
Step 1.3.2
Apply the distributive property.
Step 1.3.3
Simplify the numerator.
Step 1.3.3.1
Simplify each term.
Step 1.3.3.1.1
Multiply by .
Step 1.3.3.1.2
Multiply by by adding the exponents.
Step 1.3.3.1.2.1
Move .
Step 1.3.3.1.2.2
Multiply by .
Step 1.3.3.1.3
Multiply by .
Step 1.3.3.2
Subtract from .
Step 1.3.4
Reorder terms.
Step 1.3.5
Factor out of .
Step 1.3.5.1
Factor out of .
Step 1.3.5.2
Factor out of .
Step 1.3.5.3
Factor out of .
Step 2
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
Multiply the exponents in .
Step 2.3.1
Apply the power rule and multiply exponents, .
Step 2.3.2
Multiply by .
Step 2.4
Differentiate using the Product Rule which states that is where and .
Step 2.5
Differentiate.
Step 2.5.1
By the Sum Rule, the derivative of with respect to is .
Step 2.5.2
Differentiate using the Power Rule which states that is where .
Step 2.5.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.5.4
Simplify the expression.
Step 2.5.4.1
Add and .
Step 2.5.4.2
Multiply by .
Step 2.5.5
Differentiate using the Power Rule which states that is where .
Step 2.5.6
Simplify by adding terms.
Step 2.5.6.1
Multiply by .
Step 2.5.6.2
Add and .
Step 2.6
Multiply by by adding the exponents.
Step 2.6.1
Multiply by .
Step 2.6.1.1
Raise to the power of .
Step 2.6.1.2
Use the power rule to combine exponents.
Step 2.6.2
Add and .
Step 2.7
Differentiate using the chain rule, which states that is where and .
Step 2.7.1
To apply the Chain Rule, set as .
Step 2.7.2
Differentiate using the Power Rule which states that is where .
Step 2.7.3
Replace all occurrences of with .
Step 2.8
Simplify with factoring out.
Step 2.8.1
Multiply by .
Step 2.8.2
Factor out of .
Step 2.8.2.1
Factor out of .
Step 2.8.2.2
Factor out of .
Step 2.8.2.3
Factor out of .
Step 2.9
Cancel the common factors.
Step 2.9.1
Factor out of .
Step 2.9.2
Cancel the common factor.
Step 2.9.3
Rewrite the expression.
Step 2.10
By the Sum Rule, the derivative of with respect to is .
Step 2.11
Since is constant with respect to , the derivative of with respect to is .
Step 2.12
Differentiate using the Power Rule which states that is where .
Step 2.13
Multiply by .
Step 2.14
Since is constant with respect to , the derivative of with respect to is .
Step 2.15
Combine fractions.
Step 2.15.1
Add and .
Step 2.15.2
Multiply by .
Step 2.15.3
Combine and .
Step 2.16
Simplify.
Step 2.16.1
Apply the distributive property.
Step 2.16.2
Apply the distributive property.
Step 2.16.3
Simplify the numerator.
Step 2.16.3.1
Simplify each term.
Step 2.16.3.1.1
Rewrite as .
Step 2.16.3.1.2
Expand using the FOIL Method.
Step 2.16.3.1.2.1
Apply the distributive property.
Step 2.16.3.1.2.2
Apply the distributive property.
Step 2.16.3.1.2.3
Apply the distributive property.
Step 2.16.3.1.3
Simplify and combine like terms.
Step 2.16.3.1.3.1
Simplify each term.
Step 2.16.3.1.3.1.1
Rewrite using the commutative property of multiplication.
Step 2.16.3.1.3.1.2
Multiply by by adding the exponents.
Step 2.16.3.1.3.1.2.1
Move .
Step 2.16.3.1.3.1.2.2
Multiply by .
Step 2.16.3.1.3.1.3
Multiply by .
Step 2.16.3.1.3.1.4
Multiply by .
Step 2.16.3.1.3.1.5
Multiply by .
Step 2.16.3.1.3.1.6
Multiply by .
Step 2.16.3.1.3.2
Add and .
Step 2.16.3.1.4
Apply the distributive property.
Step 2.16.3.1.5
Simplify.
Step 2.16.3.1.5.1
Multiply by .
Step 2.16.3.1.5.2
Multiply by .
Step 2.16.3.1.5.3
Multiply by .
Step 2.16.3.1.6
Multiply by by adding the exponents.
Step 2.16.3.1.6.1
Move .
Step 2.16.3.1.6.2
Multiply by .
Step 2.16.3.1.7
Multiply by .
Step 2.16.3.1.8
Multiply by .
Step 2.16.3.1.9
Multiply by .
Step 2.16.3.2
Combine the opposite terms in .
Step 2.16.3.2.1
Subtract from .
Step 2.16.3.2.2
Add and .
Step 2.16.3.2.3
Subtract from .
Step 2.16.3.2.4
Add and .
Step 3
The second derivative of with respect to is .