What is the second derivative of $(f \cdot g) (x)$ $(f * g)(x)$ 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

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

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

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