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 \square
(round to two decimal places as needed.)

Explanation:

Step1: Substitute into the derivative formula

Given \(\frac{d}{dx}6^{x}=m6^{x}\) with \(m = 1.792\). For part (a), when \(x = 5\), we substitute \(x\) into the formula \(\frac{d}{dx}(6^{x})\big|_{x = 5}=m\times6^{5}\).

Step2: Calculate the value

We know that \(6^{5}=6\times6\times6\times6\times6 = 7776\). Then \(m\times6^{5}=1.792\times7776\).

$$1.792\times7776=(1 + 0.792)\times7776=7776+0.792\times7776$$

$$0.792\times7776=(0.8 - 0.008)\times7776=0.8\times7776-0.008\times7776$$

$$0.8\times7776 = 6220.8$$

,

$$0.008\times7776=62.208$$

$$0.792\times7776=6220.8- 62.208=6158.592$$

$$7776+6158.592=13934.592\approx13934.59$$

Answer:

\(13934.59\)