# Use Logarithmic Differentiation to Find the Derivative y=(x^2+2)^2(x^4+4)^4 Let , take the natural logarithm of both sides .
Expand the right hand side.
Rewrite as .
Expand by moving outside the logarithm.
Expand by moving outside the logarithm.
Differentiate the expression using the chain rule, keeping in mind that is a function of .
Differentiate the left hand side using the chain rule.
Differentiate the right hand side.
Differentiate .
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Combine and .
Combine and .
Combine and .
Multiply by .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Combine and .
Combine and .
Combine and .
Multiply by .
Combine terms.
To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Multiply and .
Multiply and .
Reorder the factors of .
Combine the numerators over the common denominator.
Isolate and substitute the original function for in the right hand side.
Simplify the right hand side.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify each term.
Apply the distributive property.
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Multiply by .
Apply the distributive property.
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Multiply by .
Apply the distributive property.
Expand by multiplying each term in the first expression by each term in the second expression.
Simplify each term.
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Rewrite using the commutative property of multiplication.
Multiply by .
Multiply by by adding the exponents.
Move .
Multiply by .
Raise to the power of .
Use the power rule to combine exponents.
Rewrite using the commutative property of multiplication.
Multiply by .
Multiply by by adding the exponents.
Move .
Use the power rule to combine exponents.
Rewrite using the commutative property of multiplication.
Multiply by .     