if $f(x)=(6 - ln(x))^{6}$, determine $f(1).$ $f(1)=$
if $f(x)=(6 - ln(x))^{6}$, determine $f(1).$ $f(1)=$
Answer
Explanation:
Step1: Apply chain - rule
Let $u = 6-\ln(x)$, then $f(x)=u^{6}$. The chain - rule states that $f^\prime(x)=\frac{df}{du}\cdot\frac{du}{dx}$. First, find $\frac{df}{du}$: $\frac{df}{du}=6u^{5}=6(6 - \ln(x))^{5}$. Second, find $\frac{du}{dx}$: $\frac{du}{dx}=-\frac{1}{x}$. So, $f^\prime(x)=6(6 - \ln(x))^{5}\cdot(-\frac{1}{x})=-\frac{6(6 - \ln(x))^{5}}{x}$.
Step2: Evaluate $f^\prime(1)$
Substitute $x = 1$ into $f^\prime(x)$. Since $\ln(1)=0$, then $f^\prime(1)=-\frac{6(6-\ln(1))^{5}}{1}=-\frac{6(6 - 0)^{5}}{1}=-6\times6^{5}=-6^{6}=-46656$.
Answer:
$-46656$