My brother is preparing for the university and asked me the following multiple choice question.
$$\frac{d}{dx}(x^3 * e^{x^2})$$
- a) $e^{x^2}*x^2*(1+2x)$
- b) $e^{x^2}*x^2*(3+2x)$
- c) $e^{x^2}*x^2*(3+2x^2)$
- d) $e^{x^2}*x^2*(3-2x)$
- e) $e^{x^2}*x^2*(3-2x^2)$
Even though I find $e^{x^2}*x^2*(3+2x^2)$, the answer is $e^{x^2}*x^2*(3+2x)$. I wonder where do I make the mistake. What I did is as follows:
By product rule:
$$(x^3 * e^{x^2})' \Rightarrow 3x^2 * e^{x^2} + x^3 * (e^{x^2})'$$ Since $(e^{x^2})' = 2x * e^{x^2}$, the equation becomes $$3x^2 * e^{x^2} + x^3 * 2x * e^{x^2}$$ $$e^{x^2} * x^2 * (3 + 2x^2)$$
Thanks
I'm writing this just to show a possibility.
If $e^{x^2}$ represents $(e^x)^2$ in the textbook, which should be written as $e^{2x}$, then the answer is $(e^x)^2\cdot x^2\cdot (3+2x)$.
(By the way, since $e^{x^2}$ means $e^{(x^2)}$ in general, your calculation has no mistakes.)