Practice Problems

1. $\ln (5 x)$

3. $\ln \left(x^{k}\right)$

5. $\ln (\cos (x))$

7. $\log _{2} 5 \mathrm{x}$

9. $\ln (\sin (x))$

11. $\log _{2}(\sin (x))$

13. $\log _{5} 5^{x}$

15. $x \cdot \ln (3 x)$

17. $\frac{\ln (x)}{x}$

19. $\ln (\sqrt{5 x-3})$

21. $\frac{\mathbf{d}}{\mathbf{d} \mathbf{w}} \cos (\ln (\mathrm{w}))$

23. $\frac{\mathbf{d}}{\mathbf{d t}} \ln (\sqrt{t+1})$

25. $\mathrm{D}\left(5^{\sin (\mathrm{x})}\right)$

27. $\frac{\mathbf{d}}{\mathbf{d x}} \ln (\sec (\mathrm{x})+\tan (\mathrm{x}))$

29. Find a point $\mathrm{P}$ on the graph of $\mathrm{f}(\mathrm{x})=\ln (\mathrm{x})$ so the tangent line to $\mathrm{f}$ at $\mathrm{P}$ goes through the origin.

31. Rumor. The percent of a population, $\mathrm{p}(\mathrm{t})$, who have heard a rumor by time $\mathrm{t}$ is often modeled $p(t)=\frac{100}{1+A e^{-t}}=100\left(1+A e^{-t}\right)^{-1}$ for some positive constant A. Calculate how fast the rumor is spreading, $\frac{\mathbf{d} \mathrm{p}(\mathrm{t})}{\mathbf{d t}}$.

In problems 33 – 41, find a function with the given derivative. 

33. $\mathrm{f}^{\prime}(\mathrm{x})=\frac{8}{\mathrm{x}}$

35. $f^{\prime}(x)=\frac{\cos (x)}{3+\sin (x)}$

37. $\mathrm{g}^{\prime}(\mathrm{x})=3 \mathrm{e}^{5 \mathrm{x}}$

39. $f^{\prime}(x)=2 x \cdot e^{\left(x^{2}\right)}$

41. $h^{\prime}(x)=\frac{\cos (x)}{\sin (x)}$

Problems 43 – 47 involve parametric equations.

43. At time $t$ minutes, robot $A$ is at $(t, 2 t+1)$ and robot $B$ is at $\left(t^{2}, 2 t^{2}+1\right)$.

(a) Where is each robot when $\mathrm{t}=0$ and $\mathrm{t}=1$ ?

(b) Sketch the path each robot follows during the first minute.

(c) Find the slope of the tangent line, $\mathrm{dy} / \mathrm{dx}$, to the path of each robot at $\mathrm{t}=1$ minute.

(d) Find the speed of each robot at $\mathrm{t}=1$ minute.

(e) Discuss the motion of a robot which follows the path $(\sin (\mathrm{t}), 2 \sin (\mathrm{t})+1)$ for 20 minutes.

45. For the parametric graph in Fig. 9, determine whether $\frac{d x}{d t}, \frac{d y}{d t}$ and $\frac{d y}{d x}$ are positive, negative or zero when $\mathrm{t}=1$ and $\mathrm{t}=3$.

47. $x(t)=R \cdot(t-\sin (t)), y(t)=R \cdot(1-\cos (t))$. (a) Graph $(x(t), y(t))$ for $0 \leq t \leq 4 \pi$.

(b) Find $\mathrm{dx} / \mathrm{dt}, \mathrm{dy} / \mathrm{dt}$, the tangent slope $\mathrm{dy} / \mathrm{dx}$, and speed when $\mathrm{t}=\pi / 2$ and $\pi$.

(The graph of $(\mathrm{x}(\mathrm{t}), \mathrm{y}(\mathrm{t}))$ is called a cycloid and is the path of a light attached to the edge of a rolling wheel with radius $\mathrm{R}$).

49. Describe the path of a robot whose location at time $\mathrm{t}$ is

(a) $(3 \cdot \cos (\mathrm{t}), 5 \cdot \sin (\mathrm{t}))$

(b) $(\mathrm{A} \cdot \cos (\mathrm{t}), \mathrm{B} \cdot \sin (\mathrm{t}))$

(c) Give the parametric equations so the robot will move along the same path as in part (a) but in the opposite direction.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-3.6-Some-Applications-of-the-Chain-Rule.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. $\mathbf{D}(\ln (5 \mathrm{x}))=\frac{1}{5 \mathrm{x}} 5=\frac{1}{\mathrm{x}}$

3. $\mathbf{D}\left(\ln \left(\mathrm{x}^{\mathrm{k}}\right)\right)=\frac{1}{\mathrm{x}} \mathrm{k} \quad \mathrm{k} \mathrm{x}^{\mathrm{k}-1}=\frac{\mathrm{k}}{\mathrm{x}}$

5. $\quad \mathbf{D}(\ln (\cos (x)))=\frac{1}{\cos (x)}(-\sin (x))=-\tan (x)$

7. $\quad \mathbf{D}\left(\log _{2}(5 \mathrm{x})\right)=\frac{1}{5 \mathrm{x} \ln (2)}(5)=\frac{1}{\mathrm{x} \ln (2)}$

9. $\quad \mathbf{D}(\ln (\sin (x)))=\frac{1}{\sin (x)}(\cos (x))=\cot (x)$

11. $\quad \mathbf{D}\left(\log _{2}(\sin (x))\right)=\frac{1}{\sin (x)} \frac{1}{\ln (2)}(\cos (x))=\frac{\cot (x)}{\ln (2)}$

13. $\quad \mathbf{D}\left(\log _{5}\left(5^{\mathrm{x}}\right)\right)=\mathbf{D}(\mathrm{x})=1$

15. $\quad D(x \ln (3 x))=x \cdot \frac{1}{3 x} \cdot 3+\ln (3 x)=1+\ln (3 x)$

17. $\quad \mathrm{D}\left(\frac{\ln (\mathrm{x})}{\mathrm{x}}\right)=\frac{\mathrm{x} \cdot \frac{1}{\mathrm{x}}-\ln (\mathrm{x}) \cdot 1}{\mathrm{x}^{2}}=\frac{1-\ln (\mathrm{x})}{\mathrm{x}^{2}}$

19. $\mathbf{D}\left(\ln \left((5 \mathrm{x}-3)^{1 / 2}\right)\right)=\frac{1}{(5 \mathrm{x}-3)^{1 / 2}} \cdot \mathrm{D}\left((5 \mathrm{x}-3)^{1 / 2}\right)=\frac{1}{(5 \mathrm{x}-3)^{1 / 2}} \cdot \frac{1}{2} \cdot(5 \mathrm{x}-3)^{-1 / 2} \cdot \mathrm{D}(5 \mathrm{x}-3)=\frac{5}{2} \cdot \frac{1}{5 \mathrm{x}-3}$

21. $\frac{\mathrm{d}}{\mathrm{dw}}(\cos (\ln (\mathrm{w})))=\{-\sin (\ln (\mathrm{w}))\} \frac{1}{\mathrm{w}}=\frac{-\sin (\ln (\mathrm{w}))}{\mathrm{w}}$

23. $\quad \frac{\mathrm{d}}{\mathrm{dt}}(\ln (\sqrt{\mathrm{t}+1}))=\frac{1}{2(\mathrm{t}+1)}$

25. $\quad \mathbf{D}\left(5^{\sin (x)}\right)=5^{\sin (x)} \ln (5) \cos (x)$

$27.$ $\frac{\mathrm{d}}{\mathrm{dx}} \ln (\sec (\mathrm{x})+\tan (\mathrm{x}))=\frac{1}{\sec (\mathrm{x})+\tan (\mathrm{x})}\left(\sec (\mathrm{x}) \tan (\mathrm{x})+\sec ^{2}(\mathrm{x})\right)=\sec (\mathrm{x})$

29. $\mathrm{f}(\mathrm{x})=\ln (\mathrm{x}), \mathrm{f}^{\prime}(\mathrm{x})=\frac{1}{\mathrm{x}}$. Let $\mathrm{P}=(\mathrm{p}, \ln (\mathrm{p}))$. Then we must satisfy $\mathrm{y}-\ln (\mathrm{p})=\frac{1}{\mathrm{p}}(\mathrm{x}-\mathrm{p})$ with $\mathrm{x}=0$ and $y=0:-\ln (p)=-1$ so $p=e$ and $P=(e, 1)$.

31. $p(t)=100\left(1+A e^{-t}\right)^{-1} \cdot \frac{\text { d }}{\text { dt }} p(t)=100(-1)\left(1+A e^{-t}\right)^{-2}\left(A e^{-t}(-1)\right)=\frac{100 \mathrm{Ae}^{-t}}{\left(1+A e^{-t}\right)^{2}}$.

33. $f(x)=8 \ln (x)+$ any constant

35. $f(x)=\ln (3+\sin (x))+$ any constant

37. $g(x)=\frac{3}{5} e^{5 x}+$ any constant

39. $f(x)=e^{x^{2}}+$ any constant

41. $\mathrm{h}(\mathrm{x})=\ln (\sin (\mathrm{x}))+$ any constant

43. A: $(t, 2 t+1), B:\left(t^{2}, 2 t^{2}+1\right)$

45. When $\mathrm{t}=1, \mathrm{dx} / \mathrm{dt}=+, \mathrm{dy} / \mathrm{d} \mathrm{t}=-, \mathrm{dy} / \mathrm{d} \mathrm{x}=-.$ WHen $\mathrm{t}=3, \mathrm{dx} / \mathrm{dt}=-, \mathrm{dy} / \mathrm{dt}=-, \mathrm{dy} / \mathrm{d} \mathrm{x}=+$.

47. $\begin{aligned} &x(t)=R(t-\sin (t)) \\ &y(t)=R(1-\cos (t)) \end{aligned}$

(a) graph

(b) $\mathrm{dx} / \mathrm{dt}=\mathrm{R}(1-\cos (\mathrm{t})), \mathrm{dy} / \mathrm{dt}=\mathrm{R} \sin (\mathrm{t})$, so $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sin (\mathrm{t})}{1-\cos (\mathrm{t})}$.

When $t=\pi / 2$, then $d x / d t=R, d y / d t=R$ so $d y / d x=1$ and speed $=\sqrt{R^{2}+R^{2}}=R \sqrt{2}$

When $\mathrm{t}=\pi, \mathrm{dx} / \mathrm{dt}=2 \mathrm{R}, \mathrm{dy} / \mathrm{dt}=0$ so $\mathrm{dy} / \mathrm{d} \mathrm{x}=0$ and speed $=\sqrt{(2 \mathrm{R})^{2}+0}=2 \mathrm{R}$.

49. (a) The ellipse $\left(\frac{x}{3}\right)^{2}+\left(\frac{y}{5}\right)^{2}=1$.

(b) The ellipse $\left(\frac{x}{A}\right)^{2}+\left(\frac{y}{B}\right)^{2}=1$ if $A \neq 0$ and $B \neq 0$

(c) $(3 \cdot \cos (\mathrm{t}),-5 \cdot \sin (\mathrm{t}))$ works.

If $A=0$, the motion is oscillatory along the $x$-axis. If $\mathrm{B}=0$, the motion is oscillatory along the $\mathrm{y}$-axis.