Question

compute the following derivatives using the formulas below.

$m = \frac{d}{dx}(6^{x})|_{x = 0} \approx 1.792$

$\frac{d}{dx}6^{x} = m6^{x}$

(a) $\frac{d}{dx}(6^{x})|_{x = 5}$ (b) $\frac{d}{dx}(6^{x})|_{x = -3}$

(a) $\frac{d}{dx}(6^{x})|_{x = 5} \approx 13934.59$
(round to two decimal places as needed.)

(b) $\frac{d}{dx}(6^{x})|_{x = -3} \approx \square$
(round to two decimal places as needed.)

Explanation:

Step1: Substitute into the formula

Given \(\frac{d}{dx}6^{x}=m6^{x}\) and \(m = 1.792\). For part (b), when \(x=-3\), we substitute \(x=-3\) into the formula \(\frac{d}{dx}(6^{x})\big|_{x = - 3}=m\times6^{-3}\).

Step2: Calculate the value

Since \(6^{-3}=\frac{1}{6^{3}}=\frac{1}{216}\), then \(m\times6^{-3}=1.792\times\frac{1}{216}\).

$$1.792\times\frac{1}{216}=\frac{1.792}{216}\approx0.01$$

Answer:

\(0.01\)