Question

find the second derivative of f.
f(x) =

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:

\(f^{\prime\prime}(x)=\frac{50}{x^{3}}\)