graciewarren1498 graciewarren1498

03-10-2017
Mathematics

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If f(x) = f(x)g(x), where f and g have derivatives of all orders, show that f'' = f ''g + 2f 'g' + fg''.

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LammettHash LammettHash

03-10-2017

Differentiating once, we have

[tex]f'(x)=f'(x)g(x)+f(x)g'(x)[/tex]

Differentiating again,

[tex]f''(x)=f''(x)g(x)+f'(x)g'(x)+f'(x)g'(x)+f(x)g''(x)[/tex]
[tex]f''(x)=f''(x)g(x)+2f'(x)g'(x)+f(x)g''(x)[/tex]

as needed.

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