Question

Use the Product Rule twice to find a formula (fg)'' in terms of f and g and their first and second derivatives.

Derivatives
ANSWERED
asked 2021-05-30
Use the Product Rule twice to find a formula (fg)'' in terms of f and g and their first and second derivatives.

Answers (1)

2021-05-31
Step 1
Product rule in derivative:
[A(x).B(x)]' = A(x).B'(x) + A'(x).B(x)
Step 2
To find (fg)'' = ?
(fg)'' = [(fg)']'
= [f.g' + f'.g]'
= [(f.g') + (f'.g)]'
= (f.g')' + (f'.g)'
= (f.g'' + f'.g') + (f'.g' + f''.g)
= f.g'' + f'.g' + f'.g' + f''.g
= f.g'' + 2f'.g' + f''.g
Step 3
(fg)'' = f.g'' + 2f'.g' + f''.g
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