double derivative of ln(x^5) — step-by-step solution

Let's differentiate:

{(\ln (x^{5}))}^{'}

We differentiate using:

{(\ln (f))}^{'} = {(f)}^{'} \cdot \frac{1}{f}

(where f =

x^{5}

)

{(x^{5})}^{'} \cdot \frac{1}{x^{5}}

\frac{5 x^{4}}{x^{5}}

\frac{5}{x}

To simplify differentiation, we rewrite

\frac{5}{x}

5 x^{-1}

Let's differentiate:

{(5 x^{-1})}^{'}

We differentiate using:

{(f^{n})}^{'} = n \cdot {(f)}^{'} \cdot f^{(n - 1)}

(where f =

x

)

5 \cdot -1 \cdot x^{-2}

-5 x^{-2}

▷Answer: $- \frac{5}{x^{2}}$