MA005: Calculus I

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.