Let , take the natural logarithm of both sides .

Rewrite as .

Expand by moving outside the logarithm.

Expand by moving outside the logarithm.

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 .

Add and .

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 .

Add and .

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.

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.

Add and .

Multiply by .

Apply the distributive property.

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Add and .

Multiply by .

Add and .

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.

Add and .

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.

Add and .

Rewrite using the commutative property of multiplication.

Multiply by .

Multiply by by adding the exponents.

Move .

Use the power rule to combine exponents.

Add and .

Rewrite using the commutative property of multiplication.

Multiply by .

Add and .

Add and .

Use Logarithmic Differentiation to Find the Derivative y=(x^2+2)^2(x^4+4)^4