Question

estimate $f(3)$ for $f(x)=6^{x}$. be sure your answer is accurate to within 0.1 of the actual value.
$f(3)approx$
be sure that you can explain your reasoning.

Explanation:

Step1: Recall derivative formula

The derivative of $y = a^{x}$ is $y'=a^{x}\ln(a)$. For $f(x)=6^{x}$, its derivative $f'(x)=6^{x}\ln(6)$.

Step2: Evaluate at $x = 3$

Substitute $x = 3$ into $f'(x)$. So $f'(3)=6^{3}\ln(6)$.
We know that $6^{3}=216$ and $\ln(6)\approx1.79176$. Then $f'(3)=216\times1.79176$.
$f'(3)=216\times1.79176 = 387.02016\approx387.0$.

Answer:

$387.0$