Q. 0 - Calculus

Question

Problem Zero: Read the section and make your own summary of the material.

Step-by-Step Solution

Answer

Rules for calculating basic derivatives

Step 1. Given Information

$$\begin{gathered} \mathrm{Real} \mathrm{number} : \mathrm{k}, \mathrm{m}, \mathrm{b}. \\ \mathrm{Functions} : \mathrm{f}\left(\mathrm{x}\right), \mathrm{g}\left(\mathrm{x}\right) \end{gathered}$$

Step 2. Summary

$$\begin{gathered} \mathrm{For} \mathrm{real} \mathrm{number} \mathrm{k}, \mathrm{m}, \mathrm{b} : \\ \frac{d}{d\mathrm{x}}(\mathrm{mx} + \mathrm{b}) = \mathrm{m} \\ \frac{d}{d\mathrm{x}}\left({\mathrm{x}}^{\mathrm{k}}\right) = \mathrm{k}.{\mathrm{x}}^{\mathrm{k}-1} \\ \mathrm{Constant} \mathrm{multiple} \mathrm{rule} : \frac{d}{d\mathrm{x}}(\mathrm{k}.\mathrm{f}(\mathrm{x}\left)\right) = \mathrm{k}.\mathrm{f}\text{'}\left(\mathrm{x}\right) \\ \mathrm{Sum} \mathrm{rule} : \frac{d}{d\mathrm{x}}\left(\mathrm{f}\right(\mathrm{x}) + \mathrm{g}(\mathrm{x}\left)\right) = \mathrm{f}\text{'}\left(\mathrm{x}\right) + \mathrm{g}\text{'}\left(\mathrm{x}\right) \\ \mathrm{Difference} \mathrm{rule} : \frac{d}{d\mathrm{x}}\left(\mathrm{f}\right(\mathrm{x}) - \mathrm{g}(\mathrm{x}\left)\right) = \mathrm{f}\text{'}\left(\mathrm{x}\right) - \mathrm{g}\text{'}\left(\mathrm{x}\right) \\ \mathrm{Product} \mathrm{rule}: \frac{d}{d\mathrm{x}}\left(\mathrm{f}\right(\mathrm{x}).\mathrm{g}(\mathrm{x}\left)\right) = \mathrm{f}\left(\mathrm{x}\right).\mathrm{g}\text{'}\left(\mathrm{x}\right) + \mathrm{f}\text{'}\left(\mathrm{x}\right).\mathrm{g}\left(\mathrm{x}\right) \\ \mathrm{Quotient} \mathrm{rule} : \frac{d}{d\mathrm{x}}\left(\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}\right) = \frac{\mathrm{g}\left(\mathrm{x}\right).\mathrm{f}\text{'}\left(\mathrm{x}\right) - \mathrm{f}\left(\mathrm{x}\right).\mathrm{g}\text{'}\left(\mathrm{x}\right)}{{\left(\mathrm{g}\left(\mathrm{x}\right)\right)}^2} \end{gathered}$$