What is the second derivative of #(f * g)(x)# if f and g are functions such that #f'(x)=g(x)# and #g'(x)=f(x)#?

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Answer 1 · 2018-03-19T13:40:39

#(4f*g)(x)#

Let #P(x) = (f*g)(x) = f(x)g(x)#

Then using the product rule:

#P'(x) = f'(x)g(x)+f(x)g'(x)#.

Using the condition given in the question, we get:

#P'(x) = (g(x))^2+(f(x))^2#

Now using the power and chain rules:

#P''(x) = 2g(x)g'(x) + 2f(x)f'(x)#.

Applying the special condition of this question again, we write:

#P''(x) = 2g(x)f(x)+2f(x)g(x) = 4f(x)g(x) = 4(f*g)(x)#

Answer 2 · 2018-03-19T14:42:32

Another answer in case #f*g# is meant to be the composition of #f# and #g#

We want to find the second derivative of #(f*g)(x)=f(g(x))#

We differentiate once using the chain rule.

#d/dxf(g(x))=f'(g(x))g'(x)=f'(g(x))f(x)#

Then we differentiate again using the product chain rules

#d/dxf'(g(x))f(x)=f''(g(x))g'(x)f(x)+f'(x)f'(g(x))#

#=f''(g(x))[f(x)]^2+g(x)f'(g(x))#