Derivatives of Trigonometric Functions

1. $\frac{dy}{dx}=2x- \sec x \tan x$
2. $\frac{dy}{dx}=2x \cot x-x^2 \csc^2 x$
3. $\frac{dy}{dx}=\frac{x \sec x \tan x- \sec x}{x^2}$
4. $\frac{dy}{dx}=(1- \sin x)(1- \sin x)- \cos x(x+ \cos x)$
5. $\frac{dy}{dx}=\frac{2 \csc^2 x}{(1+ \cot x)^2}$

6. $y=−x$

   

7. $y=x+\frac{2-3\pi}{2}$

   

8. $y=−x$

   

9. $\frac{d^2 y}{dx^2} = 3 \cos x-x \sin x$
10. $\frac{d^2 y}{dx^2} = \frac{1}{2} \sin x$
11. $\frac{d^2 y}{dx^2} = 2\csc x( \csc^2 x + \cot^2 x)$
12. $x = \frac{(2n+1)\pi}{4}$, where $n$ is an integer
13. $(\frac{\pi}{4},1), \, (\frac{3\pi}{4},-1)$
14. $a=0, \, b=3$
15. $y^{\prime}=5 \cos (x)$, increasing on $(0,\frac{\pi}{2}), \, (\frac{3\pi}{2},\frac{5\pi}{2})$, and $(\frac{7\pi}{2},12)$
16. $\frac{d^3 y}{dx^3} = 3 \sin x$
17. $\frac{d^4 y}{dx^4} = 5 \cos x$
18. $\frac{d^3 y}{dx^3} = 720x^7-5 \tan (x) \sec^3 (x)- \tan^3 (x) \sec (x)$

The Chain Rule

1. $\frac{dy}{dx} = 18u^2 \cdot 7=18(7x-4)^2 \cdot 7$
2. $\frac{dy}{dx} = −\sin u \cdot \frac{-1}{8}=−\sin (\frac{−x}{8}) \cdot \frac{-1}{8}$
3. $\frac{dy}{dx} = \frac{8x-24}{2\sqrt{4u+3}}=\frac{4x-12}{\sqrt{4x^2-24x+3}}$
4. 1. $f(u) = u^3, \, u=3x^2+1$
   2. $\frac{dy}{dx} = 18x(3x^2+1)^2$
5. 1. $f(u)=u^7, \, u=\frac{x}{7}+\frac{7}{x}$
   2. $\frac{dy}{dx} = 7(\frac{x}{7}+\frac{7}{x})^6 \cdot (\frac{1}{7}-\frac{7}{x^2})$
6. 1. $f(u)= \csc u, \, u=\pi x+1$
   2. $\frac{dy}{dx} = −\pi \csc (\pi x+1) \cdot \cot (\pi x+1)$
7. 1. $f(u)=-6u^{-3}, \, u= \sin x$
   2. $\frac{dy}{dx} = 18 \sin^{-4} x \cdot \cos x$
8. $\frac{dy}{dx} = \frac{4}{(5-2x)^3}$
9. $\frac{dy}{dx} = 6(2x^3-x^2+6x+1)^2(3x^2-x+3)$
10. $\frac{dy}{dx} = -3(\tan x+ \sin x)^{-4} \cdot (\sec^2 x+ \cos x)$
11. $\frac{dy}{dx} = -7 \cos (\cos 7x) \cdot \sin 7x$
12. $\frac{dy}{dx} = -12 \cot^2 (4x+1) \cdot \csc^2 (4x+1)$
13. $10$
14. $-\frac{1}{8}$
15. –$4$
16. –$12$
17. $10\frac{3}{4}$
18. $y=-\frac{1}{2}x$
19. $x= \pm \sqrt{6}$
20. 1. $-\frac{200}{343}$ m/s
    2. $\frac{600}{2401} \, \text{m/s}^2$
    3. The train is slowing down since velocity and acceleration have opposite signs
21. 1. $C^{\prime}(x)=0.0003x^2-0.04x+3$
    2. $\frac{dC}{dt}=100 \cdot (0.0003x^2-0.04x+3)$
    3. Approximately $$90,300$ per week
22. 1. $\frac{dS}{dt}=-\frac{8\pi r^2}{(t+1)^3}$
    2. The volume is decreasing at a rate of $-\frac{\pi}{36} \, \text{ft}^3/\text{min}$.
23. $\approx 2.3$ ft/hr

Derivatives of Inverse Functions

1. 1. 
   2. $(f^{-1})^{\prime}(1) \approx 2$
2. 1. 
   2. $(f^{-1})^{\prime}(1) \approx -1/\sqrt{3}$
3. 1. $6$
   2. $x=f^{-1}(y)=(\frac{y+3}{2})^{1/3}$
   3. $\frac{1}{6}$
4. 1. $1$
   2. $x=f^{-1}(y)= \sin^{-1} y$
   3. $1$
5. $\frac{1}{5}$
6. $\frac{1}{3}$
7. $1$
8. 1. $4$
   2. $y=4x$
9. 1. $-\frac{1}{13}$
   2. $y=-\frac{1}{13}x+\frac{18}{13}$
10. $\large \frac{2x}{\sqrt{1-x^4}}$
11. $\large \frac{-1}{\sqrt{1-x^2}}$
12. $\large \frac{3(1 + \tan^{-1} x)^2}{1+x^2}$
13. $\large \frac{-1}{(1+x^2)(\tan^{-1} x)^2}$
14. $\large \frac{x}{(5-x^2)\sqrt{4-x^2}}$
15. –$1$
16. $\frac{1}{2}$
17. $\frac{1}{10}$
18. 1. $v(t)=\frac{1}{1+t^2}$
    2. $a(t)=\frac{-2t}{(1+t^2)^2}$
    3. $v(2)=0.2, \, v(4)=0.06, \, v(6)=0.03; \, a(2)=-0.16, \, a(4)=-0.028, \, a(6)=-0.0088$
    4. The hockey puck is decelerating/slowing down at $2$, $4$, and $6$ seconds.
19. –$0.0168$ radians per foot
20. 1. $\frac{d\theta}{dx}=\frac{10}{100+x^2}-\frac{40}{1600+x^2}$
    2. $\frac{18}{325}, \, \frac{9}{340}, \, \frac{42}{4745}, \, 0$
    3. As a person moves farther away from the screen, the viewing angle is increasing, which implies that as he or she moves farther away, his or her screen vision is widening.
    4. $-\frac{54}{12905}, \, -\frac{3}{500}, \, -\frac{198}{29945}, \, -\frac{9}{1360}$
    5. As the person moves beyond $20$ feet from the screen, the viewing angle is decreasing. The optimal distance the person should sit for maximizing the viewing angle is $20$ feet

Implicit Differentiation

1. $\frac{dy}{dx}=\frac{-2x}{y}$
2. $\frac{dy}{dx}=\frac{x}{3y}-\frac{y}{2x}$
3. $\frac{dy}{dx}=\large \frac{y-\frac{y}{2\sqrt{x+4}}}{\sqrt{x+4}-x}$
4. $\frac{dy}{dx}=\large \frac{y^2 \cos(xy)}{2y- \sin(xy)-xy \cos xy}$
5. $\frac{dy}{dx}=\large \frac{-3x^2 y-y^3}{x^3+3xy^2}$
6. $y=-\frac{1}{2}x+2$
7. $y=\large \frac{1}{\pi +12}x-\frac{3\pi +38}{\pi +12}$
8. $y=0$
9. 1. $y=−x+2$
   2. $(3,-1)$
10. 1. $(\pm \sqrt{7},0)$
    2. Slope is $-2$ at both intercepts
    3. They are parallel since the slope is the same at both intercepts.
11. $y=−x+1$
12. 1. –$0.5926$
    2. When $$81$ is spent on labor and $$16$ is spent on capital, the amount spent on capital is decreasing by $$0.5926$ per $$1$ spent on labor.
13. $\frac{dy}{dx}=-8$
14. $\frac{dy}{dx}=-2.67$
15. $y^{\prime}=-\frac{1}{\sqrt{1-x^2}}$

Derivatives of Exponential and Logarithmic Functions

1. $f^{\prime}(x) = 2xe^x+x^2 e^x$
2. $f^{\prime}(x) = e^{x^3 \ln x}(3x^2 \ln x+x^2)$
3. $f^{\prime}(x) = \dfrac{4}{(e^x+e^{−x})^2}$
4. $f^{\prime}(x) = 2^{4x+2} \cdot \ln 2+8x$
5. $f^{\prime}(x) = \pi x^{\pi -1} \cdot \pi^x + x^{\pi} \cdot \pi^x \ln \pi$
6. $f^{\prime}(x) = \frac{5}{2(5x-7)}$
7. $f^{\prime}(x) = \frac{\tan x}{\ln 10}$
8. $f^{\prime}(x) = 2^x \cdot \ln 2 \cdot \log_3 7^{x^2-4} + 2^x \cdot \frac{2x \ln 7}{\ln 3}$
9. $\frac{dy}{dx} = (\sin 2x)^{4x} [4 \cdot \ln(\sin 2x) + 8x \cdot \cot 2x]$
10. $\frac{dy}{dx} = x^{\log_2 x} \cdot \frac{2 \ln x}{x \ln 2}$
11. $\frac{dy}{dx} = x^{\cot x} \cdot [−\csc^2 x \cdot \ln x+\frac{\cot x}{x}]$
12. $\frac{dy}{dx} = x^{-1/2}(x^2+3)^{2/3}(3x-4)^4 \cdot [\frac{-1}{2x}+\frac{4x}{3(x^2+3)}+\frac{12}{3x-4}]$
13. $y=\frac{-1}{5+5 \ln 5}x+(5+\frac{1}{5+5 \ln 5})$
14. 1. $x=e \approx 2.718$
    2. $y^{\prime}>0$ on $(e,\infty)$, and $y^{\prime}<0$ on $(0,e)$
15. 1. $P=500,000(1.05)^t$ individuals
    2. $P^{\prime}(t)=24395 \cdot (1.05)^t$ individuals per year
    3. $39,737$ individuals per year
16. 1. At the beginning of 1960 there were $5.3$ thousand cases of the disease in New York City. At the beginning of 1963 there were approximately $723$ cases of the disease in New York City.
    2. At the beginning of 1960 the number of cases of the disease was decreasing at rate of $-4.611$ thousand per year; at the beginning of 1963, the number of cases of the disease was decreasing at a rate of $-0.2808$ thousand per year.
17. $p=35741(1.045)^t$
18.  

19. Years since 1790	$P''$
    $0$	$69.25$
    $10$	$107.5$
    $20$	$167.0$
    $30$	$259.4$
    $40$	$402.8$
    $50$	$625.5$
    $60$	$971.4$
    $70$	$1508.5$

20.