How to create a dynamic dependent dropdown menu using ipywidgets?

You can do this by setting a if statement in your dropdown_handler function that checks if the value chosen in your independent dropdown is Dropdown_input_1 or Dropdown_input2. Still in the dropdown_handler, you can then modify accordingly the options arguments to either dependent_drop_down_elements['Dropdown_input_1'] or dependent_drop_down_elements['Dropdown_input_2'].

See code below:

import ipywidgets as widgets
from ipywidgets import HBox, Label
from ipywidgets import Layout, Button, Box, FloatText, Textarea, Dropdown, Label, IntSlider
import time
import pandas as pd

#Create DF
df = df = pd.DataFrame(columns = ['Dropdown_column', 'Float_column'])
df

# Layout
form_item_layout = Layout(
    display='flex',
    flex_flow='row',
    justify_content='space-between',
)


button_item_layout = Layout(
    display='flex',
    flex_flow='row',
    justify_content='center',
    padding = '5%'
)


# Independent dropdown item

drop_down_input = 'Dropdown_input_1'
drop_down = widgets.Dropdown(options=('Dropdown_input_1', 'Dropdown_input_2'))


# Dependent drop down
# Dependent drop down elements
dependent_drop_down_elements = {}
dependent_drop_down_elements['Dropdown_input_1'] = ['A', 'B']
dependent_drop_down_elements['Dropdown_input_2'] = ['C', 'D', 'E'] 

# Define dependent drop down

dependent_drop_down = widgets.Dropdown(options=(dependent_drop_down_elements['Dropdown_input_1']))

def dropdown_handler(change):
    global drop_down_input
    print('\r','Dropdown: ' + str(change.new),end='')
    drop_down_input = change.new
    #If statement checking on dropdown value and changing options of the dependent dropdown accordingly
    if change.new=='Dropdown_input_2':
       dependent_drop_down.options=dependent_drop_down_elements['Dropdown_input_2']
    elif change.new=='Dropdown_input_1':
        dependent_drop_down.options=dependent_drop_down_elements['Dropdown_input_1']
drop_down.observe(dropdown_handler, names='value')

# Button

button = widgets.Button(description='Add row to dataframe')
out = widgets.Output()
def on_button_clicked(b):
    global df
    button.description = 'Row added'
    time.sleep(1)
    with out:
      new_row = {'Dropdown_column': drop_down_input, 'Float_column': float_input}
      df = df.append(new_row, ignore_index=True)
      button.description = 'Add row to dataframe'
      out.clear_output()  
      display(df)
button.on_click(on_button_clicked)

# Form items

form_items = [         
    Box([Label(value='Independent dropdown'),
         drop_down], layout=form_item_layout),
    Box([Label(value='Dependent dropdown'),
         dependent_drop_down], layout=form_item_layout)
         ]

form = Box(form_items, layout=Layout(
    display='flex',
    flex_flow='column',
    border='solid 1px',
    align_items='stretch',
    width='30%',
    padding = '1%'
))
display(form)
display(out)

As an example, this is what the output looks like when you pick Dropdown_input_2 on the first widget: