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

Activity7.4Supplement

Read the next lab early!

Before coming to the next lab session, Lab 8, read the special note at the beginning of that lab. You are asked to do some reading outside of class before you begin working on the lab in class. More explanation is given in that note.

Subsection7.4.1Exercises

1

The curve \(x\fe{\sin}{xy}=y\) is shown in Figure 7.4.1.

Find a formula for \(\lz{y}{x}\) and use that formula to determine the \(x\)-coordinate at each of the two points the curve crosses the \(x\)-axis. (Note: The tangent line to the curve is vertical at each of these points.) Scales have deliberately been omitted in Figure 7.4.1.

Figure7.4.1\(x\fe{\sin}{xy}=y\)
2

For the curve \(x\fe{\sin}{xy}=y\) plotted in Figure 7.4.1, find the slope of the tangent line at \(\point{\sqrt{\sfrac{\pi}{3}}}{\sqrt{\sfrac{\pi}{12}}}\).

3

Solutions to the equation \(\fe{\ln}{x^2y^2}=x+y\) are graphed in Figure 7.4.2.

Determine the equation of the tangent line to this curve at the point \(\point{1}{-1}\).

Figure7.4.2\(\fe{\ln}{x^2y^2}=x+y\)
Hint

Use the process of logarithmic differentiation to find a first derivative formula for each of the following functions.

4

\(y=\dfrac{x\fe{\sin}{x}}{\sqrt{x-1}}\)

5

\(y=\dfrac{e^{2x}}{\fe{\sin^4}{x}\sqrt[4]{x^5}}\)

6

\(y=\dfrac{\fe{\ln}{4x^3}}{x^5\fe{\ln}{x}}\)