Question

find f(x).
f(x)=4x^{2}-12x - \frac{5}{x^{3}}
f(x)=\square

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=4x^{2}-12x - \frac{5}{x^{3}}\) as \(f(x)=4x^{2}-12x - 5x^{-3}\).

Step2: Find the first - derivative \(f^{\prime}(x)\)

Use the power rule \((x^{n})^\prime=nx^{n - 1}\).
For \(y = 4x^{2}\), \(y^\prime=4\times2x^{2-1}=8x\).
For \(y=-12x\), \(y^\prime=-12\).
For \(y = - 5x^{-3}\), \(y^\prime=-5\times(-3)x^{-3 - 1}=15x^{-4}\).
So, \(f^{\prime}(x)=8x-12 + 15x^{-4}\).

Step3: Find the second - derivative \(f^{\prime\prime}(x)\)

Again, use the power rule.
For \(y = 8x\), \(y^\prime=8\).
For \(y=-12\), \(y^\prime = 0\).
For \(y = 15x^{-4}\), \(y^\prime=15\times(-4)x^{-4 - 1}=-60x^{-5}\).

Answer:

\(f^{\prime\prime}(x)=8-\frac{60}{x^{5}}\)