Exercises

Use the General Rule, §4.7 in differentiating the following functions:

1. $y = 3x^2$

   Ans: $\frac{dy}{dx} = 6x$

2. $y = x^2 + 2$

   Ans: $\frac{dy}{dx} = 2x$

3. $y = 5 - 4x$

   Ans: $\frac{dy}{dx} = - 4$

4. $s = 2t^2 - 4$

   Ans: $\frac{ds}{dt} = 4t$

5. $y = \frac{1}{x}$

   Ans: $\frac{dy}{dx} = -\frac{1}{x^2}$

6. $y = \frac{x + 2}{x}$

   Ans: $\frac{dy}{dx} = -\frac{-2}{x^2}$

7. $y = x^3$

   Ans: $\frac{dy}{dx} = 3x^2$

8. $y = 2x^2 - 3$

   Ans: $\frac{dy}{dx} = 4x$

9. $y = 1 - 2x^3$

   Ans: $\frac{dy}{dx} = -6x^2$

10. $\rho = a\theta^2$

    Ans: $\frac{d\rho}{d\theta} = 2a\theta$

11. $y = \frac{2}{x^2}$

    Ans: $\frac{dy}{dx} = -\frac{4}{x^3}$

12. $y = \frac{3}{x^2 - 1}$

    Ans: $\frac{dy}{dx} = -\frac{6x}{(x^2 - 1)^2}$

    Here's how to use SAGE to verify this:

    [fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
    
    sage: y = 3/(x^2-1)
    sage: diff(y,x)
    -6*x/(x^4 - 2*x^2 + 1)
    

13. $y = 7x^2 + x$

14. $s = at^2 - 2bt$

15. $r = 8t + 3t^2$

16. $y = \frac{3}{x^2}$

17. $s = -\frac{a}{2t + 3}$

18. $y = bx^3 - cx$

19. $\rho = 3\theta^3 - 2\theta^2$

20. $y = \frac{3}{4}x^2 - \frac{1}{2}x$

21. $y = \frac{x^2 - 5}{x}$

22. $\rho = \frac{\theta^2}{1 + \theta}$

23. $y = \frac{1}{2}x^2 + 2x$

24. $z = 4x - 3x^2$

25. $\rho = 3\theta + \theta^2$

26. $y = \frac{ax + b}{x^2}$

27. $z = \frac{x^3 + 2}{x}$

28. $y = x^2 - 3x + 6$

    Ans: $y' = 2x - 3$

29. $s = 2t^2 + 5t - 8$

    Ans: $s' = 4t + 5$ Here's how to use SAGE to verify this (for simplicity, we set $h=\Delta t$):

    [fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
    
    sage: h = var("h")
    sage: t = var("t")
    sage: s(t) = 2*t^2 + 5*t - 8
    sage: Deltas = s(t+h)-s(t)
    sage: (Deltas/h).expand()
    4*t + 2*h + 5
    sage: limit((s(t+h)-s(t))/h,h=0)
    4*t + 5
    sage: diff(s(t),t)
    4*t + 5
    

30. $\rho = 5\theta^3 - 2\theta + 6$

    Ans: $\rho' = 15\theta^2 - 2$

31. $y = ax^2 + bx + c$

    Ans: $y' = 2ax + b$