# Standart Derivatives Table
This table contains the fundamental derivatives, which need to be known so that other functions can be derived.

| $f(x)$ | $\frac{df}{dx}$ |
|--------|-----------------|
| $k$ (constant) | $0$ |
| $x$ | $1$ |
| $x^k$ | $kx^{k-1}$ |
| $e^x$ | $e^x$ |
| $k^x$ | $k^x \ln{k}$ |
| $\sqrt{x}$ | $\frac{1}{2\sqrt{x}}$ |
| $\ln{x}$ | $\frac{1}{x}$ |
| $\log_k{x}$ | $\frac{1}{x\ln{k}}$ |
| $\sin{x}$ | $\cos{x}$ |
| $\cos{x}$ | $-\sin{x}$ |
| $\tan{x}$ | $\sec^2{x}$ |
| $\arcsin{x}$ | $\frac{1}{\sqrt{1-x^2}}$ |
| $\arccos{x}$ | $-\frac{1}{\sqrt{1-x^2}}$ |
| $\arctan{x}$ | $\frac{1}{1+x^2}$ |
| $\sinh{x}$ | $\cosh{x}$ |
| $\cosh{x}$ | $\sinh{x}$ |
| $\tanh{x}$ | $1 - \tanh^2{x}$ |
| $\text{arsinh}{x}$ | $\frac{1}{\sqrt{x^2+1}}$ |
| $\text{arcosh}{x}$ | $\frac{1}{\sqrt{x^2-1}}$ |
| $\text{artanh}{x}$ | $\frac{1}{1- x^2}$ |

# Practice with Python
## Symbolic derivatives with SymPy
This practice consists of generating this same table using the symbolic mathematical package SymPy. The first step is to import the package.


```python
from sympy import *
```

Now, let's define the variables and symbolic constants that we will use, for that we use the Symbol class.


```python
x = Symbol('x')
k = Symbol('k')
```

Now, let's define a list of functions $f(x)$.


```python
functions = [
    k,
    x,
    x**k,
    exp(x),
    k**x,
    sqrt(x),
    ln(x),
    log(x, k),
    sin(x),
    cos(x),
    tan(x),
    asin(x),
    acos(x),
    atan(x),
    sinh(x),
    cosh(x),
    tanh(x),
    asinh(x),
    acosh(x),
    atanh(x)
]
```

Now, let's generate a table of symbolic derivatives. To derive SymPy's symbolic expressions we use **diff**.


```python
for function in functions:
    derivative = diff(function, x)
    print(f'f(x) = {function} \t df/dx = {derivative}')
```

    f(x) = k 	 df/dx = 0
    f(x) = x 	 df/dx = 1
    f(x) = x**k 	 df/dx = k*x**k/x
    f(x) = exp(x) 	 df/dx = exp(x)
    f(x) = k**x 	 df/dx = k**x*log(k)
    f(x) = sqrt(x) 	 df/dx = 1/(2*sqrt(x))
    f(x) = log(x) 	 df/dx = 1/x
    f(x) = log(x)/log(k) 	 df/dx = 1/(x*log(k))
    f(x) = sin(x) 	 df/dx = cos(x)
    f(x) = cos(x) 	 df/dx = -sin(x)
    f(x) = tan(x) 	 df/dx = tan(x)**2 + 1
    f(x) = asin(x) 	 df/dx = 1/sqrt(1 - x**2)
    f(x) = acos(x) 	 df/dx = -1/sqrt(1 - x**2)
    f(x) = atan(x) 	 df/dx = 1/(x**2 + 1)
    f(x) = sinh(x) 	 df/dx = cosh(x)
    f(x) = cosh(x) 	 df/dx = sinh(x)
    f(x) = tanh(x) 	 df/dx = 1 - tanh(x)**2
    f(x) = asinh(x) 	 df/dx = 1/sqrt(x**2 + 1)
    f(x) = acosh(x) 	 df/dx = 1/sqrt(x**2 - 1)
    f(x) = atanh(x) 	 df/dx = 1/(1 - x**2)


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