Question

find the first and the second derivatives of the function.

$f(x)=x^{4}+\frac{4}{x^{6}}$

$f(x)=4x^{3}-\frac{24}{x^{7}}$

$f(x)=\square$

Explanation:

Step1: Differentiate \(4x^{3}\)

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(y = 4x^{3}\), \(y^\prime=4\times3x^{3 - 1}=12x^{2}\)

Step2: Differentiate \(-\frac{24}{x^{7}}\)

Rewrite \(-\frac{24}{x^{7}}\) as \(-24x^{-7}\). Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(y=-24x^{-7}\), \(y^\prime=-24\times(-7)x^{-7 - 1}=\frac{168}{x^{8}}\)

Answer:

\(f^{\prime\prime}(x)=12x^{2}+\frac{168}{x^{8}}\)