I got this wrong on my test, my instructor did the work on the board and used a different approach than I did, I want to know what I did wrong in my approach that caused me to get the problem wrong.

Differentiate:

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My (flawed) approach:

differentiates to:

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The correct answer is:

and I can get it the instructor's way, but not my way, but I don't understand why my way failed me.