Review 1: Further Practice

Here are a few extra practice problems.
Find the derivatives of the functions.

$y = (2 x + 1)^{5}$ $y=(2x+1)^5$
$y = {(1 - \frac{x}{7})}^{- 7}$ $y=\left(1-\frac{x}{7}\right)^{-7}$
$y = {(\frac{x^{2}}{8} + x - \frac{1}{x})}^{4}$ $y=\left(\frac{x^2}{8}+x-\frac{1}{x}\right)^4$
$y = \sec (\tan x)$ $y=\sec(\tan x)$
$y = \sin^{3} x$ $y=\sin^3 x$
$p = \sqrt{3 - t}$ $p=\sqrt{3-t}$
$y = \frac{4}{3 π} \sin 3 t + \frac{4}{5 π} \cos 5 t$ $y=\frac{4}{3\pi}\sin 3t+\frac{4}{5\pi}\cos 5t$
$s = \sin (\frac{3 π t}{2}) + \cos (\frac{3 π t}{2})$ $s=\sin\left(\frac{3\pi t}{2}\right)+\cos\left(\frac{3\pi t}{2}\right)$
$r = (\csc θ + \cot θ)^{- 1}$ $r=(\csc \theta+\cot \theta)^{-1}$
$y = x^{2} \sin^{4} x + x \cos^{- 2} x$ $y=x^2\sin^4x+x\cos^{-2}x$
$y = \frac{1}{21} (3 x - 2)^{7} + {(4 - \frac{1}{2 x^{2}})}^{- 1}$ $y=\frac{1}{21}(3x-2)^7+\left(4-\frac{1}{2x^2}\right)^{-1}$
$y = (5 - 2 x)^{_{3}} + \frac{1}{8} {(\frac{2}{x} + 1)}^{4}$ $y=(5-2x)^{_3}+\frac{1}{8}\left(\frac{2}{x}+1\right)^4$
$y = (4 x + 3)^{4} (x + 1)^{- 3}$ $y=(4x+3)^4(x+1)^{-3}$
$h (x) = x \tan (2 \sqrt{x}) + 7$ $h(x)=x\tan(2\sqrt x)+7$
$f (θ) = {(\frac{\sin θ}{1 - \cos θ})}^{2}$ $f(\theta)=\left(\frac{\sin \theta}{1-\cos \theta}\right)^2$
$r = \sin (θ^{2}) \cos (2 θ)$ $r=\sin(\theta^2)\cos(2\theta)$
$q = \sin (\frac{1}{\sqrt{t + 1}})$ $q=\sin\left(\frac{1}{\sqrt{t+1}}\right)$
$x^{2} y + x y^{2} = 6$ $x^2y+xy^2=6$
$2 x y + y^{2} = x + y$ $2xy+y^2=x+y$
$y^{2} = \frac{x - 1}{x + 1}$ $y^2=\frac{x-1}{x+1}$
$x = \tan y$ $x=\tan y$
$x + \tan (x y) = 0$ $x+\tan(xy)=0$
$y \sin (\frac{1}{y}) = 1 - x y$ $y\sin\left(\frac{1}{y}\right)=1-xy$
$x^{1 / 2} + y^{1 / 2} = 1$ $x^{1/2}+y^{1/2}=1$
$\sin (x y) = \frac{1}{2}$ $\sin(xy)=\frac{1}{2}$

Solutions:

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1. $y^{'} = 10 (2 x + 1)^{4}$ $y'=10(2x+1)^4$

2. $(- 7) (\frac{- 1}{7}) (1 - \frac{x}{7})^{- 8}$ $(-7)(\frac{-1}{7})(1-\frac{x}{7})^{-8}$

3. $y^{'} = 4 {(\frac{x^{2}}{8} + x - \frac{1}{x})}^{3} (\frac{x}{4} + 1 + \frac{1}{x^{2}})$ $y'=4\left(\frac{x^2}{8}+x-\frac{1}{x}\right)^3\left(\frac{x}{4}+1+\frac{1}{x^2}\right)$

4. $LaTeX: y'=\sec(\tan x)\tan(\tan x)\sec^2 x$ $y'=\sec(\tan x)\tan(\tan x)\sec^2 x$

5. $y^{'} = 3 \sin^{2} x \cos x$ $y'=3\sin^2 x \cos x$

6. $p^{'} = \frac{- 1}{2 \sqrt{3 - t}}$ $p'=\frac{-1}{2\sqrt{3-t}}$

7. $y^{'} = \frac{4}{π} \cos 3 t - \frac{4}{π} \sin 5 t$ $y'=\frac{4}{\pi}\cos 3t-\frac{4}{\pi}\sin 5t$

8. $s^{'} = \frac{3 π}{2} \cos (\frac{3 π t}{2}) - \frac{3 π}{2} \sin (\frac{3 π t}{2})$ $s'=\frac{3\pi}{2}\cos\left(\frac{3\pi t}{2}\right)-\frac{3\pi}{2}\sin\left(\frac{3\pi t}{2}\right)$

9. $r^{'} = (- 1) (\csc θ + \cot θ)^{- 2} (\cot θ \csc θ - \csc^{2} θ)$ $r'=(-1)(\csc \theta+\cot \theta)^{-2}(\cot \theta \csc \theta-\csc^2 \theta )$

10. $y^{'} = 2 x \sin^{4} x + 4 x^{2} \sin^{3} x \cos x + \cos^{- 2} x - 2 x \cos^{- 3} x (- \sin x)$ $y'=2x\sin^4x+4x^2\sin^3x\cos x + \cos^{-2}x-2x\cos^{-3}x(-\sin x)$

11. $y^{'} = \frac{1}{3} (3 x - 2)^{6} \cdot 3 + (- 1) {(4 - \frac{1}{2 x^{2}})}^{- 2} (x^{- 3})$ $y'=\frac{1}{3}(3x-2)^6\cdot3+(-1)\left(4-\frac{1}{2x^2}\right)^{-2}(x^{-3})$

12. $y^{'} = (- 6) (5 - 2 x)^{2} + \frac{1}{2} {(\frac{2}{x} + 1)}^{3} \frac{- 2}{x^{2}}$ $y'=(-6)(5-2x)^{2}+\frac{1}{2}\left(\frac{2}{x}+1\right)^3\frac{-2}{x^2}$

13. $y^{'} = 16 (4 x + 3)^{3} (x + 1)^{- 3} - 3 (4 x + 3)^{4} (x + 1)^{- 4}$ $y'=16(4x+3)^3(x+1)^{-3}-3(4x+3)^4(x+1)^{-4}$

14. $h^{'} (x) = x \sec^{2} (2 \sqrt{x}) x^{- \frac{1}{2}} + \tan (2 \sqrt{x})$ $h'(x)=x\sec^2(2\sqrt x)x^{-\frac{1}{2}}+\tan(2\sqrt x)$

15. $f^{'} (θ) = 2 (\frac{\sin θ}{1 - \cos θ}) (\frac{\cos θ (1 - \cos θ) - \sin^{2} θ}{(1 - \cos θ)^{2}})$ $f'(\theta)=2\left(\frac{\sin \theta}{1-\cos \theta}\right)(\frac{\cos \theta (1-\cos \theta)-\sin^2 \theta}{(1-\cos\theta)^2})$

16. $r^{'} = 2 θ \cos (θ^{2}) \cos (2 θ) + \sin (θ^{2}) (- 2) \sin (2 θ)$ $r'=2\theta\cos(\theta^2)\cos(2\theta)+\sin(\theta^2)(-2)\sin(2\theta)$

17. $q^{'} = \frac{- 1}{2} (t + 1)^{\frac{- 3}{2}} \cos (\frac{1}{\sqrt{t + 1}})$ $q'=\frac{-1}{2}(t+1)^{\frac{-3}{2}}\cos\left(\frac{1}{\sqrt{t+1}}\right)$

18. $LaTeX: y'=\frac{-y^2-2xy}{x^2+2xy}$ $y'=\frac{-y^2-2xy}{x^2+2xy}$

19. $y^{'} = \frac{2 y - 1}{1 - 2 x - 2 y}$ $y'=\frac{2y-1}{1-2x-2y}$

20. $y^{'} = \frac{1}{(x + 1)^{2} y}$ $y'=\frac{1}{(x+1)^2y}$

21. $y^{'} = \cos^{2} y$ $y'=\cos^2 y$

22. $y^{'} = \frac{- 1 - y \sec^{2} (x y)}{x \sec^{2} (x y)}$ $y'=\frac{-1-y\sec^2(xy)}{x\sec^2(xy)}$

23. $LaTeX: y'=\frac{-y}{x+\sin\frac{1}{y}-\frac{1}{y}\cos\frac{1}{y}}$ $y'=\frac{-y}{x+\sin\frac{1}{y}-\frac{1}{y}\cos\frac{1}{y}}$

24. $y^{'} = - \frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}}$ $y'=-\frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}}$

25. $y^{'} = \frac{- y}{x}$ $y'=\frac{-y}{x}$