Multiple Choice

1. If $f(x) = (x^2)(3x+1)$, find $f'(x)$.

   $6x^2 + 2x$

   $3x^2 + 2x$

   $9x^2 + 2x$

   $9x^2$

2. Find the derivative of $y = \dfrac{x+1}{x-1}$.

   $\dfrac{-2x}{(x-1)^2}$

   $\dfrac{2}{(x-1)^2}$

   $\dfrac{2x-2}{(x-1)^2}$

   $\dfrac{-2}{(x-1)^2}$

3. What is $g'(x)$ if $g(x) = (x^3 - 2)(5x + 4)$?

   $20x^3 + 12x^2 - 10$

   $5x^3 + 12x^2 - 10$

   $15x^3 + 12x^2 - 10$

   $20x^3 + 12x^2$

4. Differentiate $h(x) = \dfrac{x^2 - 3x}{x+2}$.

   $\dfrac{x^2 + 4x - 6}{(x+2)^2}$

   $\dfrac{2x - 3}{(x+2)^2}$

   $\dfrac{x^2 + 7x - 6}{(x+2)^2}$

   $\dfrac{x^2 - 2x - 6}{(x+2)^2}$

5. If $y = (4x - 1)(x^2 + 5)$, find $\dfrac{dy}{dx}$.

   $12x^2 - 2x + 20$

   $4x^2 + 20 + 2x$

   $12x^2 + 20$

   $8x^2 - 2x + 20$

6. Calculate the derivative of $f(x) = \dfrac{5}{x^2+1}$.

   $\dfrac{5}{2x}$

   $\dfrac{10}{(x^2+1)^2}$

   $\dfrac{10x}{(x^2+1)^2}$

   $\dfrac{-10x}{(x^2+1)^2}$

7. Find $f'(x)$ for $f(x) = x(x^2 - 3x + 2)$.

   $x^2 - 3x + 2$

   $3x^2 - 6x + 2$

   $2x - 3$

   $3x^2 - 3x + 2$

8. Differentiate $y = \dfrac{x^2+2x-1}{x-2}$.

   $\dfrac{x^2 - 3x - 5}{(x-2)^2}$

   $\dfrac{2x+2}{(x-2)^2}$

   $\dfrac{x^2 - 4x + 1}{(x-2)^2}$

   $\dfrac{x^2 - 4x - 3}{(x-2)^2}$

9. If $h(t) = (2t^2 + 3)(t^3 - t)$, find $h'(t)$.

   $6t^4 + 7t^2 - 3$

   $4t^4 - 4t^2$

   $10t^4 + 5t^2 - 3$

   $10t^4 + 3t^2 - 3$

10. Calculate $\dfrac{d}{dx}\!\left(\dfrac{1}{x^3 - 2}\right)$.

    $\dfrac{-3x^2}{x^3-2}$

    $\dfrac{1}{3x^2}$

    $\dfrac{-3x^2}{(x^3-2)^2}$

    $\dfrac{3x^2}{(x^3-2)^2}$

11. Find the derivative of $f(x) = (x+5)(x^2)$.

    $2x^2 + 10x$

    $2x$

    $3x^2 + 10x$

    $3x^2 + 5x$

12. What is $y'$ if $y = \dfrac{3x^2 - 2x + 1}{x^2 + 4}$?

    $\dfrac{2x^2 + 22x + 8}{(x^2+4)^2}$

    $\dfrac{4x^2 + 22x - 8}{(x^2+4)^2}$

    $\dfrac{2x^2 - 26x - 8}{(x^2+4)^2}$

    $\dfrac{2x^2 + 22x - 8}{(x^2+4)^2}$

13. Calculate $\dfrac{d}{dx}\big((x-1)(x+3)\big)$.

    $2x-2$

    $x+3$

    $2x$

    $2x+2$

14. If $g(x) = \dfrac{x^2}{x-5}$, find $g'(x)$.

    $\dfrac{x^2 - 5x}{(x-5)^2}$

    $\dfrac{2x^2 - 10x}{(x-5)^2}$

    $\dfrac{x^2 + 10x}{(x-5)^2}$

    $\dfrac{x^2 - 10x}{(x-5)^2}$

15. Find the derivative of $f(x) = x^4(2x-3)$.

    $8x^4 - 12x^3$

    $10x^4 - 12x^3$

    $10x^4 - 6x^3$

    $2x^4$

16. What is $\dfrac{dy}{dx}$ if $y = \dfrac{3x+2}{4x-1}$?

    $\dfrac{11}{(4x-1)^2}$

    $\dfrac{-11}{(4x-1)^2}$

    $\dfrac{12x-3}{(4x-1)^2}$

    $\dfrac{12x+8}{(4x-1)^2}$

17. If $f(x) = (5x^2)(x^3+x)$, find $f'(x)$.

    $25x^4 + 15x^2$

    $15x^4 + 5x^2$

    $10x^4 + 10x^2$

    $25x^4 + 10x^2$

18. What is the derivative of $y = \dfrac{x+1}{x^2+2}$?

    $\dfrac{-x^2 + 2x + 2}{(x^2+2)^2}$

    $\dfrac{-x^2 - 2x - 2}{(x^2+2)^2}$

    $\dfrac{x^2+2}{(x^2+2)^2}$

    $\dfrac{-x^2 - 2x + 2}{(x^2+2)^2}$

19. Find $f'(x)$ if $f(x) = (x^3 - 4x)(x^2 + 2x - 1)$.

    $5x^4 + 8x^3 - 15x^2 - 16x + 4$

    $5x^4 + 6x^3 - 15x^2 - 16x + 4$

    $5x^4 + 8x^3 - 15x^2 - 16x$

    $5x^4 + 8x^3 - 15x^2 - 16x + 8$

20. Calculate the derivative of $h(x) = \dfrac{x^2+3}{x^2-2}$.

    $\dfrac{-10x}{x^2-2}$

    $\dfrac{-2x}{(x^2-2)^2}$

    $\dfrac{-10x}{(x^2-2)^2}$

    $\dfrac{10x}{(x^2-2)^2}$