find the second derivative of f.\nf(x) =
find the second derivative of f.\nf(x) =
Answer
Explanation:
Step1: Find the first derivative
Use the power rule ((x^n)^\prime=nx^{n - 1}). For (y = f(x)=x+\frac{25}{x}=x + 25x^{-1}), the first derivative (f^\prime(x)=(x)^\prime+(25x^{-1})^\prime). By the power rule, ((x)^\prime = 1) and ((25x^{-1})^\prime=25\times(-1)x^{-2}). So (f^\prime(x)=1-\frac{25}{x^{2}}).
Step2: Find the second derivative
Differentiate (f^\prime(x)=1 - 25x^{-2}) with respect to (x). Using the power rule again, ((1)^\prime = 0) and ((-25x^{-2})^\prime=-25\times(-2)x^{-3}).
Answer:
(f^{\prime\prime}(x)=\frac{50}{x^{3}})