`y = (x^2 + 2)^2 (x^4 + 4)^4` Use logarithmic differentiation to find the derivative of the function.

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Chapter 3, 3.6 - Problem 39 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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mathace | (Level 3) Assistant Educator

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Given: `y=(x^2+2)^2(x^4+4)^4`      Take the logarithm of both sides of the equation, then rewrite the expression using the laws of logarithms. ``

`lny=ln[(x^2+2)^2(x^4+4)^4]`

`lny=2ln(x^2+2)+4ln(x^4+4)`

Take the derivate of both sides of the equation. 

`(1)/(y)(dy)/(dx)=(2)/(x^2+2)(2x)+(4)/(x^4+4)(4x^3)`

`(dy)/(dx)=y[(4x)/(x^2+2)+(16x^3)/(x^4+4)]`

Substitute in for y using the given. 

`(dy)/(dx)=(x^2+2)^2(x^4+4)^4[(4x)/(x^2+2)+(16x^3)/(x^4+4)]` 

The derivative is: `(4x)(x^2+2)(x^4+4)^4+(16x^3)(x^2+2)^2(x^4+4)^3`

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