(a)- if f(x) = f(x)g(x), where f and g have derivatives of all orders, show that f"=f''g+2f'g'+fg'' (b)- find simmiliar formula for f''' and f^(4) (c)-describe the pattern for higher derivatives of f.

See proof below

Step-by-step explanation:

By the rule of the derivative of a product and sum (we will omit the argument x to make clearer the calculations)

F' = (fg)' = f'g + fg'

F'' = (f'g + fg')' = (f'g)' + (fg')' = (f''g+f'g') + (f'g'+fg'') =  

f''g + 2f'g' + fg''

b) In a similar way, we can find that

F''' and are

F''' = f'''g + 3f''g' + 3 f'g'' + fg'''

c)

The pattern for higher derivatives resemble the Newton's binomial:

where

means no derivative and

are the combination of n elements taken m at a time

the correct answer is the second option

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step-by-step explanation: