For the function \( f(x)=x^{4}+\frac{2}{x} \) Find a) \( f \prime(x) \) b) \( f \prime \prime(x) \)

We start with the function   f(x) = x⁴ + (2/x). It is often easier to rewrite f(x) using exponents:   f(x) = x⁴ + 2x⁻¹. a) To find f '(x), differentiate term by term:   • The derivative of x⁴ is 4x³.   • The derivative of 2x⁻¹ is 2(–1)x^(–2) = –2x⁻². Thus,   f '(x) = 4x³ – 2x⁻², or equivalently,   f '(x) = 4x³ – 2/x². b) To find f ''(x), we differentiate f '(x):   f '(x) = 4x³ – 2x⁻². Differentiate each term:   • The derivative of 4x³ is 12x².   • The derivative of –2x⁻² is –2(–2)x^(–3) = 4x⁻³. Thus,   f ''(x) = 12x² + 4x⁻³, or equivalently,   f ''(x) = 12x² + 4/x³. So, the answers are:   a) f '(x) = 4x³ – 2/x²   b) f ''(x) = 12x² + 4/x³.