Plot functions and their guide function images into same coordinate system

Questions
,
1

In order to study characteristics such as the monotonic value range of a function, it is necessary to draw the function and its guide function into the same coordinate system.
for example:
use this code can plot the function

Plot[E^x (1 + x) + Cos[x], {x, -2, 2}]

11.png
11.png601Ă—373 3.66 KB

use such code can solve its guide function

\!\(
\*SubscriptBox[\(\[PartialD]\), \(x\)]\((
\*SuperscriptBox[\(E\), \(x\)]\ \((1 + x)\) + Cos[x])\)\)

and plot it

12.png
12.png601Ă—383 3.65 KB

but how to when i input a function then the mathematica can solve its guide function ,and last draw the two function’s images into the same coordinate. thx
[1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=11.png&userId=1828799
[2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=12.png&userId=1828799

2

Hi Lee,

Something like this?

ClearAll@f;
f[x_] := E^x (1 + x) + Cos[x]

Plot[{f[x], Evaluate@D[f[x], x]}, {x, -2, 2}, PlotLabels -> Automatic]

4040plot.png
4040plot.png720Ă—298 10.6 KB

3

Thank you! It is this effect. Further words, if the second-order guidance function of the function is to draw the same coordinate system?

4

Sure. It can be made interactive, pick the number of orders you want to show.

Manipulate[
 Plot[Evaluate@Table[D[f[x], {x, n}], {n, 0, nMax, 1}], {x, -2, 2},
  PlotLabels -> Automatic,
  PlotRange -> All,
  ImageSize -> 500],
 {{nMax, 1, "Order"}, 1, 5, 1, PopupMenu}]

8866plot.png
8866plot.png680Ă—387 17.7 KB

5

Thank you very much!
The result is beyond imagination!

However, there is still an obvious problem: the derivatives of the original function are not identified, that is, the function signature in the image does not show that the derivative function is the derivative function of the original function.
Can you display the derivative function and show which derivative function it is at the same time?
Thanks again!

6

Specifically, it shows the function analytical expression of the derivative function in the image, and at the same time, it also shows the order of the derivative function of the original function.
thx

7

If you know how to differentiate then it is quite clear from the labels which order each curve corresponds to. Both can be displayed but the labels become long and high. The code is quite messy, good practice for you to refactor it to make it cleaner. Maybe you just want to display a number corresponding to the order rather than the differential operator, you can easily change the code to do that. Or maybe add a Tooltip to show the order, there are many possibilities…

Manipulate[
 Refresh[
  functions = Table[D[f[x], {x, n}], {n, 0, nMax, 1}];
  orders = 
   Table[D[f[x], {x, n}] // Inactivate // TraditionalForm // ToString, {n, 0, nMax, 1}];
  labels = MapThread[#1 <> " = " <> ToString[#2, TraditionalForm] &, {orders, functions}];];
 Plot[functions, {x, -2, 2},
  PlotLabels -> labels,
  PlotRange -> All,
  ImageSize -> 700,
  AspectRatio -> 1,
  PlotLabel -> Row[{"f(x) = ", f[x]}]],
 {{nMax, 1, "Order"}, 1, 5, 1, PopupMenu}]

2414plot.png
2414plot.png930Ă—639 29.9 KB

8

Thank you very much. That’s what I want.