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Activity5.5The Constant Divisor Rule

When an expression is divided by the constant \(k\), we can think of the expression as being multiplied by the fraction \(\frac{1}{k}\). In this way, the constant factor rule of differentiation can be applied when a formula is multiplied or divided by a constant.

Given function \(\fe{f}{x}=\frac{x^4}{8}\) \(y=\frac{3\fe{\tan}{\alpha}}{2}\) \(z=\frac{\fe{\ln}{y}}{3}\)
You should “see” \(\fe{f}{x}=\frac{1}{8}x^4\) \(y=\frac{3}{2}\fe{\tan}{\alpha}\) \(z=\frac{1}{3}\fe{\ln}{y}\)
You should write \(\fe{\fd{f}}{x}=\frac{1}{2}x^3\) \(\lz{y}{\alpha}=\frac{3}{2}\fe{\sec^2}{\alpha}\) \(\lz{z}{y}=\frac{1}{3y}\)
Table5.5.1Examples of the Constant Divisor Rule

Subsection5.5.1Exercises

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 that you use the appropriate name for each derivative. In the last problem, \(G\), \(m_1\), and \(m_2\) are constants.

1

\(\fe{z}{t}=\dfrac{\fe{\sin^{-1}}{t}}{6}\)

2

\(\fe{V}{r}=\dfrac{\pi r^3}{3}\)

3

\(\fe{f}{r}=\dfrac{Gm_1m_2}{r^2}\)