Activity 7.4 Supplement
¶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.
2.
For the curve \(x\fe{\sin}{xy}=y\) plotted in Figure 7.4.1, find the slope of the tangent line at \(\point{\sqrt{\sfrac{\pi}{3}}}{\sqrt{\sfrac{\pi}{12}}}\text{.}\)
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 \(\point{1}{-1}\text{.}\)
It is easier to differentiate if you first use rules of logarithms to completely expand the logarithmic expression.
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.
\(y=\dfrac{\fe{\ln}{4x^3}}{x^5\fe{\ln}{x}}\)