Derivatives

• Determining Rules
• Examining Relationship between Function and Derivative

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import sympy as sy
import pandas as pd

#writing a derivative and its definition

Find the Derivative

\[f(x) = \sqrt{x} + x^3\]

Derivative at a Point

What is \(f'(1)\)?

#instantaneous rate of change of function f

#derivative as a function of x

Other Functions

• \(\sin\) and \(\cos\)
• \(e^x\)

x, h = sy.symbols('x h')

def ddf(f, x, h):
    return (f(x + h) - f(x))/h

def f(x):
    return sy.exp(x)

ddf(f, x, h)

$\displaystyle \frac{- e^{x} + e^{h + x}}{h}$

def g(x):
    return sy.sin(x)

ddf(g, x, h)

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

sy.diff(g(x))

$\displaystyle \cos{\left(x \right)}$

sy.diff(f(x))

$\displaystyle e^{x}$

Desigining a Curve

• At least 3 zeros between \(x = -2\) and \(x = 2\)
• Plot function and Derivative itself