MA005: Calculus I

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