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Activity5.4The Constant Factor Rule

The next rule you are going to practice is the constant factor rule of differentiation. This is another rule you do in your head. \begin{equation}\lzoo{x}{k\ \fe{f}{x}}=k\cdot\lzoo{x}{\fe{f}{x}}\quad\text{for }k\in\reals\text{.}\label{men-3}\tag{5.4.1}\end{equation}

In the following examples and problems several derivative rules are used that are shown in Appendix B. While working this lab you should refer to those rules; you may want to cover up the chain rule and implicit derivative columns this week! You should make a goal of having all of the basic formulas memorized within a week.

Given function \(\fe{f}{x}=6x^9\) \(\fe{f}{\theta}=2\fe{\cos}{\theta}\) \(z=2\fe{\ln}{x}\)
You should write \(\fe{\fd{f}}{x}=54x^8\) \(\fe{\fd{f}}{\theta}=-2\fe{\sin}{\theta}\) \(\lz{z}{x}=\frac{2}{x}\)
Table5.4.1Examples of the Constant Factor Rule

Subsection5.4.1Exercises

Find the first derivative formula for each of the following functions. In each case take the derivative with respect to the independent variable as implied by the expression on the right side of the equal sign. Make sure to use the appropriate name for each derivative.

1

\(z=7t^4\)

2

\(\fe{P}{x}=-7\fe{\sin}{x}\)

3

\(\fe{h}{t}=\frac{1}{3}\fe{\ln}{t}\)

4

\(\fe{z}{x}=\pi\fe{\tan}{x}\)

5

\(P=\dfrac{-8}{t^4}\)

6

\(T=4\sqrt{t}\)