Question

PRN: 20
If the functions f and g have derivability up to 2nd order, then (fg) 2 =​

Answer 1

It seems like you are searching for" 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)?

Answer:

The second derivative of  (f⋅ g)(x) $P^{\prime \prime}(x)=4(f \cdot g)(x)$.

Step-by-step explanation:

Step 1: Let $P(x)=(f \cdot g)(x)=f(x) g(x)$

Step 2: Then using the product rule:

$P^{\prime}(x)=f^{\prime}(x) g(x)+f(x) g^{\prime}(x)$

Step 3: Using the condition given in the question, we get:

$P^{\prime}(x)=(g(x))^{2}+(f(x))^{2}$

Step 4: Now using the power and chain rules:

$P^{\prime \prime}(x)=2 g(x) g^{\prime}(x)+2 f(x) f^{\prime}(x)$

Step 5: Applying the special condition of this question again, we write:

$P^{\prime \prime}(x)=2 g(x) f(x)+2 f(x) g(x)=4 f(x) g(x)=4(f \cdot g)(x)$