## Practice Problems

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

11. Find the slopes of the lines tangent to the graph in Fig. 3 at the points , and .

13. Find the slopes of the lines tangent to the graph in Fig. 4 at the points , and .

In problems 15 – 21, find using implicit differentiation and then find the slope of the line tangent to the graph of the equation at the given point.

23. Find the slope of the line tangent to the ellipse in Fig. 5 at the point .

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 in two ways: (a) by using the "usual" differentiation patterns and (b) by using logarithmic differentiation.

In problems 41–47, use logarithmic differentiation to find .

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

49. Use Table 3.

Table 3

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 to rederive the product rule: .

53. Use logarithmic differentiation on to derive a product rule for three functions: .

55. Use logarithmic differentiation to determine a pattern for the derivative of .

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

This work is licensed under a Creative Commons Attribution 3.0 License.