```python
from IPython.display import Image
from IPython.core.display import HTML 
```


```python
Image(url= "https://i.imgur.com/OKzEasd.png")
```









```python
from sympy import *
from sympy import euler
x = symbols('x')
c = symbols('c', real = True)
sympify(c).is_real
```




    True




```python
eq5 = c*E**(x)
Eq(eq5,100) #solving for c when x = 10 because the plot looks nice
```




$\displaystyle c e^{x} = 100$




```python
factorial??
```


```python
eq = x**x
Eq(Derivative(eq).doit(),E**(log(x)*x))
```




$\displaystyle x^{x} \left(\log{\left(x \right)} + 1\right) = e^{x \log{\left(x \right)}}$




```python
E**log(x)
```




$\displaystyle x$




```python
for x in range(1,10):
   print(x**x)
```

    1
    4
    27
    256
    3125
    46656
    823543
    16777216
    387420489



```python
for x in range(1,10):
    print((x**x)*(log(x)+1).evalf())
```

    1.00000000000000
    6.77258872223978
    56.6625317940390
    610.891356446692
    8154.49347635656
    130252.329796304
    2426083.68188346
    51664455.9041356
    1238670309.19442



```python
import math
x = symbols('x')
f = (x**x)*(log(x)+1)
#plot(x**x, f,yscale='log',xlim = (-1,10),line_color=1,legend=True)

p1 = plot(x**x, show=False,line_color=lambda x:math.cos(x), title="Derivatives of x**x",yscale='log',xscale='log',size = (9,5))
p2 = plot(f, show = False,line_color=lambda x:math.cos(x))
p3 = plot(diff(x**x,x,2),show=False,line_color=lambda x:math.cos(x))
p4 = plot(diff(x**x,x,3),show=False,line_color=lambda x:math.cos(x))

p5 = plot(diff(x**x,x,4),show=False,line_color=lambda x:math.cos(x))
p6 = plot(diff(x**x,x,5),show=False,line_color=lambda x:math.cos(x))
p7 = plot(diff(x**x,x,6),show=False,line_color=lambda x:math.cos(x))
p8 = plot(diff(x**x,x,7),show=False,line_color=lambda x:math.cos(x))
p9 = plot(diff(x**x,x,8),show=False,line_color=lambda x:math.cos(x))


p1.extend(p2)
p1.extend(p3)
p1.extend(p4)
p1.extend(p5)
p1.extend(p6)
p1.extend(p7)
p1.extend(p8)
p1.extend(p9)

p1.show()
```


```python
Derivative(x**x) > x**x
```




$\displaystyle \frac{d}{d x} x^{x} > x^{x}$




```python
(Derivative(x**x,x,2)> Derivative(x**x,x,1))
```




$\displaystyle \frac{d^{2}}{d x^{2}} x^{x} > \frac{d}{d x} x^{x}$




```python
(Derivative(x**x,x,2)> Derivative(x**x,x,1)).doit()
```




$\displaystyle x^{x} \left(\left(\log{\left(x \right)} + 1\right)^{2} + \frac{1}{x}\right) > x^{x} \left(\log{\left(x \right)} + 1\right)$




```python
(Derivative(x**x,x,4)> Derivative(x**x,x,3)).doit()
```




$\displaystyle x^{x} \left(\left(\log{\left(x \right)} + 1\right)^{4} + \frac{6 \left(\log{\left(x \right)} + 1\right)^{2}}{x} - \frac{4 \left(\log{\left(x \right)} + 1\right)}{x^{2}} + \frac{3}{x^{2}} + \frac{2}{x^{3}}\right) > x^{x} \left(\left(\log{\left(x \right)} + 1\right)^{3} + \frac{3 \left(\log{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right)$




```python
from sympy import oo
n = symbols('n')
Limit(Derivative(x**x,x,n),n,oo)  # this does not display what I want accurately in LaTeX
```




$\displaystyle \lim_{n \to \infty} \frac{d^{2}}{d nd x} x^{x}$




```python
log(x,x**x)
```




$\displaystyle \frac{\log{\left(x \right)}}{\log{\left(x^{x} \right)}}$




```python
for i in range(2,5):
    print(log(x,i**i))
```

    log(x)/log(4)
    log(x)/log(27)
    log(x)/log(256)



```python
Eq(Derivative(log(x,x**x)),Derivative(log(x,x**x)).doit())
```




$\displaystyle \frac{d}{d x} \frac{\log{\left(x \right)}}{\log{\left(x^{x} \right)}} = \frac{\left(- \log{\left(x \right)} - 1\right) \log{\left(x \right)}}{\log{\left(x^{x} \right)}^{2}} + \frac{1}{x \log{\left(x^{x} \right)}}$




```python

```


```python
Limit(x,x,2)
```




$\displaystyle \lim_{x \to 2^+} x$




```python
Eq(Derivative(E**(log(x)*x)),Derivative(E**(log(x)*x)).doit())
```


```python
Eq(((x**x)/(E**x)),simplify(((x**x)/(E**x))))
```


```python
((E**x)/(x**x))
```


```python
factorial(x).compare(simplify(((x**x)/(E**x))))
# -1 for smaller, 0 for equal and 1 for greater
```


```python
(factorial(x) > ((x**x)/(E**x))).simplify()
```


```python
plot(x**x,Derivative(eq).doit(),yscale='log') # after plotting this I noticed that it would be nice if this y= 100 right when x = 10
                 # the final solution may be useful for conceptualizing percentage of exponential growths
```


```python
log(10**4).evalf()
```


```python
Image(url= "https://i.imgur.com/F2uZGSM.png")
```


```python
eq11 = Eq(c*E**(10),100) 
eq11
# This is what I want to solve. 
```


```python
eq1 = 0.00453999208646404*exp(x)
Eq(eq1,eq1.subs(x,10).evalf())
```


```python
eq2 = 0.00454*exp(x)
Eq(eq2,eq2.subs(x,10).evalf())
```


```python
p0 = plot(eq1,eq2,show = False,xlim = (0,10),size = (12,5),legend = True,autoscale = True)
p0[0].line_color = 'r'
p0[1].line_color = 'b'
p0.show()
```


```python
s = solve(eq11)[0]
s # one possible solution for real number c when x = 10
```


```python
s.evalf()
```


```python
final_solution = (s*E**x)
phi = symbols('phi')
Eq(phi,final_solution)
```


```python
final_solution.subs(x,10)
```


```python
final_equation = s.evalf()*E**x
Eq(final_equation,100)
```


```python
(0.00453999297624849*E**10).evalf()
```


```python
p0 = plot(eq1,eq2,final_solution,show = False,xlim = (0,10),size = (12,5),legend = True,autoscale = True)
p0[0].line_color = 'r'
p0[1].line_color = 'b'
p0[2].line_color = 'g'
p0.show()
```


```python
eq14 = Eq(0.00453999297624849*E**(x),100) 
solve(eq14)
```


```python
0.00453999208646404 < 0.00453999297624849 < 0.00454
```

e69d79cafc907ea639aa272faeaa2c4b4ce0db8c
ipynb
Jupyter Notebook
Personal_Projects/Extra_Derivative_Stuff.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Personal_Projects/Extra_Derivative_Stuff.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Personal_Projects/Extra_Derivative_Stuff.ipynb
NSC9/Sample_of_Work
8f8160fbf0aa4fd514d4a5046668a194997aade6
MIT
Qwen/Qwen-72B
1. YES
2. YES