[48] how to simplify derivative?
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07-16-2023, 08:53 AM
Post: #2
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RE: [48] how to simplify derivative?
I assume that you are familiar with calculating the derivative on the HP-48G:
\( \begin{matrix} \frac{\mathrm{d} }{\mathrm{d} x} \frac{\sin(x))}{1 + \cos(x)} = \frac{\cos(x)}{1 + \cos(x)}-\frac{\sin(x) \left(-\sin(x)\right)}{\left(1 + \cos(x)\right)^2} \\ \end{matrix} \) We have to cheat a bit and use the following identities: \( \begin{matrix} \sin(x)^2 + \cos(x)^2 &=& 1 \\ \left(1 + \cos(x)\right)^2 &=& \left(1 + \cos(x)\right) \cdot \left(1 + \cos(x)\right) \end{matrix} \) Therefore we put everything on the stack for the later substitutions: 3: 1 2: '(1+COS(X))*(1+COS(X))' 1: 'COS(X)/(1+COS(X))-SIN(X)*-SIN(X)/(1+COS(X))^2' Now we can enter the equation editor by pressing the ▼ key. Select the expression \(\left(1 + \cos(x)\right)^2\) and use REPL to replace it with \(\left(1 + \cos(x)\right) \cdot \left(1 + \cos(x)\right)\). Select this whole expression: \( \begin{matrix} \frac{\sin(x) \left(-\sin(x)\right)}{\left(1 + \cos(x)\right) \cdot \left(1 + \cos(x)\right)} \end{matrix} \) Use ←A to associate left which leaves us with: \( \begin{matrix} \frac{\cos(x)}{1 + \cos(x)}-\frac{\frac{\sin(x) \left(-\sin(x)\right)}{\left(1 + \cos(x)\right)}}{\left(1 + \cos(x)\right)} \end{matrix} \) Use ←→ to commute the arguments: \( \begin{matrix} \frac{\cos(x)}{1 + \cos(x)}-\text{inv}\left(1 + \cos(x)\right) \frac{\sin(x) \left(-\sin(x)\right)}{\left(1 + \cos(x)\right)} \end{matrix} \) We do the same on the left expression as well: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \cos(x) -\text{inv}\left(1 + \cos(x)\right) \cdot \frac{\sin(x) \left(-\sin(x)\right)}{\left(1 + \cos(x)\right)} \end{matrix} \) This allows using ←M to merge-factors-left: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \left(\cos(x) - \frac{\sin(x) \left(-\sin(x)\right)}{\left(1 + \cos(x)\right)}\right) \end{matrix} \) Now we can use AF to add the fractions: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \frac{\left(\left(1 + \cos(x)\right) \cdot \cos(x) - \sin(x) \left(-\sin(x)\right)\right)}{\left(1 + \cos(x)\right)} \end{matrix} \) We use ←D to distribute-left the expression: \(\left(1 + \cos(x)\right) \cdot \cos(x)\) to end up with: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \frac{1 \cdot \cos(x) + \cos(x) \cdot \cos(x) - \sin(x) \left(-\sin(x)\right)}{\left(1 + \cos(x)\right)} \end{matrix} \) With COLCT we can clean that up a bit: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \frac{\cos(x)^2 + \sin(x)^2 + \cos(x)}{\left(1 + \cos(x)\right)} \end{matrix} \) Now we substitute \(\cos(x)^2 + \sin(x)^2\) by \(1\) using the REPL command again: \( \begin{matrix} \text{inv}\left(1 + \cos(x)\right) \cdot \frac{1 + \cos(x)}{\left(1 + \cos(x)\right)} \end{matrix} \) We can now use COLCT on the fraction and then on the whole expression and end up with: \( \begin{matrix} \frac{1}{1 + \cos(x)} \end{matrix} \) The result is then: 1: '1/(1+COS(X))' Conclusion The equation writer is severely limited:
This makes it close to useless. |
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Messages In This Thread |
[48] how to simplify derivative? - grbrum - 07-15-2023, 09:11 PM
RE: [48] how to simplify derivative? - Thomas Klemm - 07-16-2023 08:53 AM
RE: [48] how to simplify derivative? - Gilles - 07-16-2023, 10:46 AM
RE: [48] how to simplify derivative? - grbrum - 07-16-2023, 09:57 PM
RE: [48] how to simplify derivative? - BruceH - 07-18-2023, 07:54 PM
RE: [48] how to simplify derivative? - FLISZT - 07-18-2023, 08:35 PM
RE: [48] how to simplify derivative? - Gilles - 07-17-2023, 07:44 AM
RE: [48] how to simplify derivative? - Thomas Klemm - 07-19-2023, 12:59 AM
RE: [48] how to simplify derivative? - BruceH - 07-19-2023, 11:33 PM
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