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\(\newcommand{\Z}{\mathbb{Z}} \newcommand{\reals}{\mathbb{R}} \newcommand{\real}[1]{\mathbb{R}^{#1}} \newcommand{\fe}[2]{#1\mathopen{}\left(#2\right)\mathclose{}} \newcommand{\cinterval}[2]{\left[#1,#2\right]} \newcommand{\ointerval}[2]{\left(#1,#2\right)} \newcommand{\cointerval}[2]{\left[\left.#1,#2\right)\right.} \newcommand{\ocinterval}[2]{\left(\left.#1,#2\right]\right.} \newcommand{\point}[2]{\left(#1,#2\right)} \newcommand{\fd}[1]{#1'} \newcommand{\sd}[1]{#1''} \newcommand{\td}[1]{#1'''} \newcommand{\lz}[2]{\frac{d#1}{d#2}} \newcommand{\lzn}[3]{\frac{d^{#1}#2}{d#3^{#1}}} \newcommand{\lzo}[1]{\frac{d}{d#1}} \newcommand{\lzoo}[2]{{\frac{d}{d#1}}{\left(#2\right)}} \newcommand{\lzon}[2]{\frac{d^{#1}}{d#2^{#1}}} \newcommand{\lzoa}[3]{\left.{\frac{d#1}{d#2}}\right|_{#3}} \newcommand{\abs}[1]{\left|#1\right|} \newcommand{\sech}{\operatorname{sech}} \newcommand{\csch}{\operatorname{csch}} \newcommand{\lt}{ < } \newcommand{\gt}{ > } \newcommand{\amp}{ & } \)

Activity5.2The Derivative Operator

\(\lz{y}{x}\) is the name of a derivative in the same way that \(\fe{\fd{f}}{x}\) is the name of a derivative. We need a different symbol that tells us to take the derivative of a given expression (in the same way that we have symbols that tell us to take a square root, sine, or logarithm of an expression).

The symbol \(\lzo{x}\) is used to tell us to take the derivative with respect to \(x\) of something. The symbol itself is an incomplete phrase in the same way that the symbol \(\sqrt{\phantom{x}}\) is an incomplete phrase; in both cases we need to indicate the object to be manipulated—what number or formula are we taking the square root of? What number or formula are we differentiating?

One thing you can do to help you remember the difference between the symbols \(\lz{y}{x}\) and \(\lzo{x}\) is to get in the habit of always writing grouping symbols after \(\lzo{x}\). In this way the symbols \(\lzoo{x}{\fe{\sin}{x}}\) mean “the derivative with respect to \(x\) of the sine of \(x\).” Similarly, the symbols \(\lzoo{t}{t^2}\) mean “the derivative with respect to \(t\) of \(t\)-squared.”

Subsection5.2.1Exercises

Write the Leibniz notation for each of the following expressions.

1

The derivative with respect to \(\beta\) of \(\fe{\cos}{\beta}\).

2

The derivative with respect to \(x\) of \(\lz{y}{x}\).

3

The derivative with respect to \(t\) of \(\fe{\ln}{x}\). (Yes, we will do such things.)

4

The derivative of \(z\) with respect to \(x\).

5

The derivative with respect to \(t\) of \(\fe{g}{8}\). (Yes, we will also do such things.)