In problems 1-19 find \mathrm{dy} / \mathrm{dx} in two ways: (a) by differentiating implicitly and (b) by explicitly solving for \mathrm{y} and then differentiating. Then find the value of \mathrm{dy} / \mathrm{dx} at the given point using your results from both the implicit and the explicit differentiation.

1. x^{2}+y^{2}=100, point (6,8)


3. x^{2}-3 x y+7 y=5, point (2,1)


5. \frac{x^{2}}{9}+\frac{y^{2}}{16}=1, point (0,4)


7. \ln (\mathrm{y})+3 \mathrm{x}-7=0, point (2, \mathrm{e})


9. x^{2}-y^{2}=16, point (5,-3)


11. Find the slopes of the lines tangent to the graph in Fig. 3 at the points (3,1),(3,3), and (4,2).



13. Find the slopes of the lines tangent to the graph in Fig. 4 at the points ((5,0),(5,6), and (-4,3).



In problems 15 – 21, find \mathrm{dy} / \mathrm{dx} using implicit differentiation and then find the slope of the line tangent to the graph of the equation at the given point.

15. y^{3}-5 y=5 x^{2}+7, point (1,3)


17. y^{2}+\sin (y)=2 x-6, point (3,0)


19. \mathrm{e}^{\mathrm{y}}+\sin (\mathrm{y})=\mathrm{x}^{2}-3, point (2,0)


21. x^{2 / 3}+y^{2 / 3}=5, point (8,1)


23. Find the slope of the line tangent to the ellipse in Fig. 5 at the point (1,2).



25. Find \mathrm{y}^{\prime} for y=A x^{2}+B x+C and for x=A y^{2}+B y+C.


27. Find \mathrm{y}^{\prime} for A x^{2}+B x y+C y^{2}+D x+E y+F=0.


29. Find the coordinates of point A where the tangent line to the ellipse in Fig. 5 is horizontal.


31. Find the coordinates of points C and D on the ellipse in Fig. 5.


In problems 33-39 find \mathrm{dy} / \mathrm{dx} in two ways: (a) by using the "usual" differentiation patterns and (b) by using logarithmic differentiation.

33. y=\left(x^{2}+5\right)^{7} \cdot\left(x^{3}-1\right)^{4}


35. y=x^{5} \cdot(3 x+2)^{4}


37. y=e^{\sin (x)}


39. y=\sqrt{25-x^{2}}


In problems 41–47, use logarithmic differentiation to find \mathrm{dy} / \mathrm{dx}.

41. y=x^{\cos (x)}


43. y=x^{4} \cdot(x-2)^{7} \cdot \sin (3 x)


45. y=(3+\sin (x))^{x}


In problems 47-49, use the values in each table to calculate the values of the derivative in the last column.

47. Use Table 1.

Table 1

\begin{array}{c|c|c|c|c} \mathrm{x} & {\mathrm{f}(\mathrm{x})} & \ln (\mathrm{f}(\mathrm{x})) & \mathrm{D}(\ln (\mathrm{f}(\mathrm{x}))) & \mathrm{f}^{\prime}(\mathrm{x}) \\ \hline 1 & 1 & 0 & 1.2 & \\ 2 & 9 & 2.2 & 1.8 & \\ 3 & 64 & 4.2 & 2.1 & \end{array}


49. Use Table 3.

Table 3

\begin{array}{c|c|c|c|c} x & {\mathrm{f}(\mathrm{x})} & \ln (\mathrm{f}(\mathrm{x})) & \mathrm{D}(\ln (\mathrm{f}(\mathrm{x}))) & \mathrm{f}^{\prime}(\mathrm{x}) \\ \hline 1 & 5 & 1.6 & -1 & \\ 2 & 2 & 0.7 & 0 & \\ 3 & 7 & 1.9 & 2 & \end{array}


Problems 51–55 illustrate how logarithmic differentiation can be used to verify some differentiation patterns we already know (51 and 52) and to derive some new patterns (53 – 55). Assume that all of the functions are differentiable and that the function combinations are defined.

51. Use logarithmic differentiation on f \cdot g to rederive the product rule: D(f \cdot g)=f \cdot g^{\prime}+g \cdot f^{\prime}.


53. Use logarithmic differentiation on \mathrm{f} \cdot \mathrm{g} \cdot \mathrm{h} to derive a product rule for three functions: \mathbf{D}(\mathrm{f} \cdot \mathrm{g} \cdot \mathrm{h}).


55. Use logarithmic differentiation to determine a pattern for the derivative of \mathrm{f}^{\mathrm{g}}: \mathrm{D}\left(\mathrm{f}^{\mathrm{g}}\right).

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.10-Implicit-and-Logarithmic-Differentiation.pdf
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