```python
import sympy as sm
x = sm.symbols('x')
```

# Derivatives
> ## $ \frac{d}{dx}(\frac{1+sinn\,x}{1-cos\,x})^2 $


```python
sm.diff((1+sm.sin(x)) / (1 - sm.cos(x))**2,x)
```




$\displaystyle \frac{\cos{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} - \frac{2 \left(\sin{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{3}}$



>  ## $
\frac{d}{dx}(log_5(x))^{x/2}
$


```python
sm.diff(sm.log(x,5)**(x/2),x)
```




$\displaystyle \left(\frac{\log{\left(x \right)}}{\log{\left(5 \right)}}\right)^{\frac{x}{2}} \left(\frac{\log{\left(\frac{\log{\left(x \right)}}{\log{\left(5 \right)}} \right)}}{2} + \frac{1}{2 \log{\left(x \right)}}\right)$



> ## $ 
\frac{d}{dx}f(x+g(x)) \\
c.f \quad \xi ,\, \Xi \to xi
$


```python
f,g = sm.symbols('f g',cls=sm.Function)
g = g(x)
f = f(x + g)
sm.diff(f,x)
```




$\displaystyle \left(\frac{d}{d x} g{\left(x \right)} + 1\right) \left. \frac{d}{d \xi_{1}} f{\left(\xi_{1} \right)} \right|_{\substack{ \xi_{1}=x + g{\left(x \right)} }}$



# Basic AntiDerivative
> ## $ \int csc(x)cot(x)dx$


```python
sm.integrate(sm.csc(x)*sm.cot(x),x)
```




$\displaystyle - \frac{1}{\sin{\left(x \right)}}$



> ## $
\int 4sec(3x)tan(3x)dx
$


```python
sm.integrate(4*sm.sec(3*x)*sm.tan(3*x),x)
```




$\displaystyle \frac{4}{3 \cos{\left(3 x \right)}}$




```python

```

d917d2aa95b74d73ddd5ff26f80c1b17c0e1f4bb
ipynb
Jupyter Notebook
python/Vectors/derivatives.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
python/Vectors/derivatives.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
python/Vectors/derivatives.ipynb
karng87/nasm_game
a97fdb09459efffc561d2122058c348c93f1dc87
MIT
Qwen/Qwen-72B
1. YES
2. YES