Question

find the first and the second derivatives of the function.
$f(x)=x^{4}+\frac{4}{x^{6}}$

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=x^{4}+\frac{4}{x^{6}} \) as \( f(x)=x^{4}+4x^{-6} \).

Step2: Find the first - derivative using the power rule

The power rule is \( (x^{n})^\prime = nx^{n - 1} \).
For \( y = x^{4} \), \( y^\prime=4x^{4 - 1}=4x^{3} \).
For \( y = 4x^{-6} \), \( y^\prime=4\times(-6)x^{-6 - 1}=-24x^{-7} \).
So \( f^\prime(x)=4x^{3}-24x^{-7}=4x^{3}-\frac{24}{x^{7}} \).

Answer:

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