How to find nth derivative

Question

Can anyone help with this question?

How to find the nth derivative of $$ e^x(2x+3)^3? $$

For example $\exp(x)$ and $e^x$ will be displayed as $\exp(x)$ and $e^x$. — Axel, Jun 21 '21 at 08:20
@Axel I updated the question I hope it clear now... I want the solution to the question above I hope you can help with it. Thank you — Rahul, Jun 21 '21 at 11:07
@10209 I updated the question I hope it clear now... I want the solution to the question above I hope you can help with it. Thank you — Rahul, Jun 21 '21 at 11:07
@Rahul Okay so $f(x) = e^x (2x+3)^3$? I am quite confused when you write afterwards $f(x) = x e^{2x}$. — Axel, Jun 21 '21 at 11:25
@Axel Please ignore the below fx contents.... Those were not the solution I removed those lines I want to solve that question I don't know how to solve it. — Rahul, Jun 21 '21 at 11:29
@Axel I want to solve this using derivative like $dy/dx$ or $f'x$ or anything else but not in $\sum$ — Rahul, Jun 21 '21 at 11:39

Axel · Accepted Answer · 2021-06-21T12:23:29.090

-1

I hope this answer can help you.

Let $g : x \mapsto (2x+3)^3$ and $h: x \mapsto e^x$. Hence $f = gh$.

You know that for all $n \in \mathbb{N}$ you have $h^{(n)} = h$. Moreover, $g^{(k)} =0 $ for $k \geq 4$.

So using Leibniz rule,

$$f^{(n)} = \left(g+ng'+\binom{n}{2}g''+\binom{n}{3} g^{(3)}\right)h$$

$$f^{(n)}(x)=\left((2x+3)^3+6\binom{n}{1}(2x+3)^2+24\binom{n}{2}(2x+3)+48\binom{n}{3}\right)e^x$$

Hence,

$$f^{(n)}(x)=((2x+3)^3+6n(2x+3)^2+12n(n-1)(2x+3)+8n(n-1)(n-2))e^x$$

If you are not convinced, you can try to prove this by induction.

edited Jun 21 '21 at 12:23

answered Jun 21 '21 at 12:06

Axel

I accepted your answer I don't know why you downvoted but I not did anything and if you can solve this in a different way if you have them, please show it :) @Axel – Rahul Jun 21 '21 at 12:30
@Rahul This is the most efficient way I have, or you can just compute the first derivatives then try to show your result by induction... Yes I got downvoted and the question got closed. If you post a question in the future, please do your best to include your research effort and try to post a clean post using MathJax. – Axel Jun 21 '21 at 12:44
Ok I will keep this in my mind and sorry for your downvoted :( and Thank you so much – Rahul Jun 21 '21 at 12:46
No problem, there are some quality standards on this website, if you want to post later I suggest you to check them. And just out of curiosity why did you take $f(x) = x e^{2x}$ in your first edits? – Axel Jun 21 '21 at 13:04
I just copy that from the Internet solution. Can it solve it using this $$ f(x), f'(x), f''(x) $$ – Rahul Jun 21 '21 at 13:16
Ok, yes that does not help. It would have been better if you really tried to calculate the first derivatives of $(2x+3)^3 e^x$ by yourself and did put them in your post. Well, you know what to do in the future. Best of luck :-) – Axel Jun 21 '21 at 13:32

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