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Calculus Examples
Step 1
Since is constant with respect to , the derivative of with respect to is .
Step 2
Differentiate using the Quotient Rule which states that is where and .
Step 3
Step 3.1
Apply the power rule and multiply exponents, .
Step 3.2
Multiply by .
Step 4
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
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
Simplify the expression.
Step 5.5.1
Add and .
Step 5.5.2
Multiply by .
Step 6
Multiply by .
Step 7
Step 7.1
Combine.
Step 7.2
Apply the distributive property.
Step 7.3
Cancel the common factor of .
Step 7.3.1
Cancel the common factor.
Step 7.3.2
Rewrite the expression.
Step 8
Differentiate using the Power Rule which states that is where .
Step 9
Step 9.1
Multiply by .
Step 9.2
Combine and .
Step 10
Step 10.1
Apply the distributive property.
Step 10.2
Apply the distributive property.
Step 10.3
Simplify the numerator.
Step 10.3.1
Simplify each term.
Step 10.3.1.1
Simplify by moving inside the logarithm.
Step 10.3.1.2
Apply the distributive property.
Step 10.3.1.3
Rewrite using the commutative property of multiplication.
Step 10.3.1.4
Multiply .
Step 10.3.1.4.1
Multiply by .
Step 10.3.1.4.2
Reorder and .
Step 10.3.1.4.3
Simplify by moving inside the logarithm.
Step 10.3.1.5
Simplify each term.
Step 10.3.1.5.1
Multiply by by adding the exponents.
Step 10.3.1.5.1.1
Move .
Step 10.3.1.5.1.2
Multiply by .
Step 10.3.1.5.2
Multiply the exponents in .
Step 10.3.1.5.2.1
Apply the power rule and multiply exponents, .
Step 10.3.1.5.2.2
Multiply by .
Step 10.3.1.6
Apply the distributive property.
Step 10.3.1.7
Multiply .
Step 10.3.1.7.1
Multiply by .
Step 10.3.1.7.2
Simplify by moving inside the logarithm.
Step 10.3.1.8
Multiply .
Step 10.3.1.8.1
Multiply by .
Step 10.3.1.8.2
Reorder and .
Step 10.3.1.8.3
Simplify by moving inside the logarithm.
Step 10.3.1.9
Simplify each term.
Step 10.3.1.9.1
Rewrite using the commutative property of multiplication.
Step 10.3.1.9.2
Multiply the exponents in .
Step 10.3.1.9.2.1
Apply the power rule and multiply exponents, .
Step 10.3.1.9.2.2
Multiply by .
Step 10.3.1.9.3
Multiply the exponents in .
Step 10.3.1.9.3.1
Apply the power rule and multiply exponents, .
Step 10.3.1.9.3.2
Multiply by .
Step 10.3.2
Reorder factors in .
Step 10.4
Multiply by by adding the exponents.
Step 10.4.1
Multiply by .
Step 10.4.1.1
Raise to the power of .
Step 10.4.1.2
Use the power rule to combine exponents.
Step 10.4.2
Add and .
Step 10.5
Factor out of .
Step 10.5.1
Factor out of .
Step 10.5.2
Factor out of .
Step 10.5.3
Factor out of .
Step 10.5.4
Factor out of .
Step 10.5.5
Factor out of .
Step 10.6
Factor out of .
Step 10.6.1
Factor out of .
Step 10.6.2
Factor out of .
Step 10.6.3
Factor out of .
Step 10.7
Expand by moving outside the logarithm.
Step 10.8
Expand by moving outside the logarithm.
Step 10.9
Cancel the common factors.
Step 10.9.1
Factor out of .
Step 10.9.2
Cancel the common factor.
Step 10.9.3
Rewrite the expression.
Step 10.10
Simplify the numerator.
Step 10.10.1
Multiply by .
Step 10.10.2
Multiply by .
Step 10.10.3
Factor out of .
Step 10.10.3.1
Factor out of .
Step 10.10.3.2
Factor out of .
Step 10.10.3.3
Factor out of .
Step 10.10.3.4
Factor out of .
Step 10.10.3.5
Factor out of .