Question

If $f\left(x\right)={\left(\ln x\right)}^3$
a. Write $f\left(x^z\right)$ in terms of $f\left(x\right)$
b. From $\left(a\right)$ deduce $f\left(x^n\right)$ in terms of $kf\left(x\right)$
where $k$ is perfect cube given $n$ is a positive integer
c. Find the derivative of $f\left(x^n\right)$ at $x=n$

Answer 1

If $f\left(x\right)=(lnx{)}^3$

$f\left(x^z\right)=\left(ln\right(n^z){)}^3$

$f\left(x^z\right)=\left(zln\right(x){)}^3$

$f\left(x^z\right)=3^{3}f\left(x\right)$

$f\left(x^n\right)=n^{3}f\left(x\right)$

$n^3=K$

$f\left(x^n\right)=Kf\left(x\right)$ [where $K=n^3$]

$\frac{d}{dn}\left(f\right(n^n\left)\right)$ at $x=n$

$=\frac{d}{dn}\left[Kf\right(n\left)\right]x=4$

$=K\frac{d}{dn}f\left(x\right)=K{f}^{\prime }\left(n\right)|x=n$

$=K{f}^{\prime }\left(n\right)$

$=n^3{f}^{\prime }\left(n\right)$

$=n^3\left(3ln\right(x{)}^2)\frac{1}{n}$

$=3(lnn{)}^2{n}^2$