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I've got a question asking me to "differentiate $x^3 e^{-2x}$ using the product rule.

So I differentiate using it $(u v)'=u'v+uv'$ and get

  • $u: x^3$
  • $u':3x^2$
  • $v: e^{-2x}$
  • $v':-2e^{-2x}$

Adding them together: $x^3$ $-2e^{-2x}$ + $3x^2$ $e^{-2x}$

The answer I'm 'supposed' to get, however, is $dy/dx= e^{-2x}(3x^2-2x^3)$

What's the logic from going to the final answer, from what I got before? It looked like simple factorization at first, but the logic behind it isn't too clear to me.

Any insights?

Thanks!

5 Answers 5