To differentiate a quotient of two functions, you can apply the quotient rule. The quotient rule states that for two functions f(x) and g(x), the derivative of \frac{f(x)}{g(x)} is given by the formula:

\left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2}

To solve for the derivative of a quotient, follow these steps:

1. Identify the functions f(x) and g(x) in the quotient \frac{f(x)}{g(x)}.

2. Find the derivative of f(x) and g(x), denoted as f'(x) and g'(x) respectively.

3. Apply the quotient rule formula: \left(\frac{f(x)}{g(x)}\right)' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{(g(x))^2}, plugging in the values of f'(x), g'(x), f(x), and g(x) into the formula.

4. Simplify the expression to obtain the derivative of the quotient.

5. If necessary, further simplify or rearrange the expression to make it more readable or usable in a specific context.

1. Find the derivative of f(x) = \frac{2x^2 + 3x + 4}{x^2 + x - 1}.

Solution:

To find the derivative of the quotient, f(x) = \frac{g(x)}{h(x)}, we can use the quotient rule:

f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}.

First, find g'(x) and h'(x):

g'(x) = 4x + 3 and h'(x) = 2x + 1.

Now, plug the values into the formula:

f'(x) = \frac{(4x+3)(x^2 + x - 1) - (2x^2 + 3x + 4)(2x+1)}{(x^2 + x - 1)^2}.

After simplifying, f'(x) = \frac{4x^3 + 4x^2 + 4x - 3 - 4x^3 - 2x^2 - 8x - 1}{(x^2 + x - 1)^2}.

f'(x) = \frac{2x^2 - 4x - 4}{(x^2 + x - 1)^2}.

2. Determine the derivative of f(x) = \frac{3x^3 + 2x^2 + x - 5}{x^2 - 4x + 4}.
Solution:

Using the quotient rule:

f'(x) = \frac{(9x^2 + 4x + 1)(x^2 - 4x + 4) - (3x^3 + 2x^2 + x - 5)(2x - 4)}{(x^2 - 4x + 4)^2}.

After simplifying, f'(x) = \frac{9x^4 - 36x^3 + 36x^2 + 4x^3 - 16x^2 + 16x + x^2 - 4x - 4 - 6x^4 - 4x^3 - 2x^2 + 10x}{(x^2 - 4x + 4)^2}.

f'(x) = \frac{3x^4 - 16x^3 + 22x^2 + 22x - 4}{(x^2 - 4x + 4)^2}.

3. Find the derivative of f(x) = \frac{x^4 + x^3 + x^2 - x + 1}{x^2 - 1}.

Solution:

Using the quotient rule:

f'(x) = \frac{(4x^3 + 3x^2 + 2x - 1)(x^2 - 1) - (x^4 + x^3 + x^2 - x + 1)(2x)}{(x^2 - 1)^2}.

After simplifying, f'(x) = \frac{4x^5 - 4x^3 + 3x^4 - 3x^2 + 2x^3 - 2x - x^5 - x^4 - x^3 + x + 2x}{(x^2 - 1)^2}.

f'(x) = \frac{3x^5 + 2x^4 + x^3 - x}{(x^2 - 1)^2}.

4. Calculate the derivative of f(x) = \frac{5x^3 - 2x^2 + x + 3}{x^2 + 3x + 2}.

Solution:

Using the quotient rule:
f'(x) = \frac{(15x^2 - 4x + 1)(x^2 + 3x + 2) - (5x^3 - 2x^2 + x + 3)(2x + 3)}{(x^2 + 3x + 2)^2}.
After simplifying, f'(x) = \frac{15x^4 + 45x^3 + 10x^3 + 30x^2 + x^2 + 3x - 4x^3 - 12x^2 - 8x - 5x^3 + 2x^2 - x - 3}{(x^2 + 3x + 2)^2}.
f'(x) = \frac{15x^4 + 40x^3 + 20x^2 - 5x - 3}{(x^2 + 3x + 2)^2}.

5. Find the derivative of f(x) = \frac{x^3 + x^2 - x - 1}{x^2 - 2x + 1}.

Solution:

Using the quotient rule:

f'(x) = \frac{(3x^2 + 2x - 1)(x^2 - 2x + 1) - (x^3 + x^2 - x - 1)(2x - 2)}{(x^2 - 2x + 1)^2}.

After simplifying, f'(x) = \frac{3x^4 - 6x^3 + 2x^3 - 4x^2 + x^2 + 2x - x - 2 - 2x^3 + 4x^2 - 2x + 2}{(x^2 - 2x + 1)^2}.

f'(x) = \frac{3x^4 - 4x^3 + 5x^2 + 3}{(x^2 - 2x + 1)^2}.

Questions

1. Find the derivative of the function  f(x) = (x^2 + 3x) / (2x + 4).
f'(x) = \frac{2x(2x+4) - (x^2 + 3x)2}{(2x + 4)^2}

2. Determine the derivative of g(x) = (2x^3 - x^2) / (3x - 1).
g'(x) = \frac{6x^2(3x - 1) - (2x^3 - x^2)3}{(3x - 1)^2}

3. Calculate the derivative of h(x) = (4x^2 + 5x) / (x^2 + 2x).
h'(x) = \frac{8x(x^2 + 2x) - (4x^2 + 5x)(2x + 2)}{(x^2 + 2x)^2}

4. Find the derivative of the function k(x) = (3x^2 + 2x) / (4x^2 - 2x).
k'(x) = \frac{6x(4x^2 - 2x) - (3x^2 + 2x)8}{(4x^2 - 2x)^2}

5. Determine the derivative of l(x) = (5x^3 + 4x) / (x^3 - 3x).
l'(x) = \frac{15x^2(x^3-3x) - (5x^3 + 4x)(3x^2 - 3)}{(x^3 - 3x)^2}

6. Calculate the derivative of m(x) = (2x^2 - x) / (x^2 + x).
m'(x) = \frac{4x(x^2 + x) - (2x^2 - x)(2x + 1)}{(x^2 + x)^2}

7. Find the derivative of the function n(x) = (x^2 + 3x) / (2x^2 - x).
n'(x) = \frac{2x(2x^2 - x) - (x^2 + 3x)(4x - 1)}{(2x^2 - x)^2}

8. Determine the derivative of p(x) = (4x^3 - 2x) / (x^3 + 2x).
p'(x) = \frac{12x^2(x^3 + 2x) - (4x^3 - 2x)(3x^2 + 2)}{(x^3 + 2x)^2}

9. Calculate the derivative of q(x) = (3x^2 - x) / (4x^2 + 5x).
q'(x) = \frac{6x(4x^2 + 5x) - (3x^2 - x)(8x + 5)}{(4x^2 + 5x)^2}

10. Find the derivative of the function r(x) = (x^3 + 2x) / (3x^3 - x).
r'(x) = \frac{3x^2(3x^3 - x) - (x^3 + 2x)(9x^2 - 1)}{(3x^3 - x)^2}

11. Determine the derivative of s(x) = (4x^2 - 3x) / (2x^2 + 4x).
s'(x) = \frac{8x(2x^2 + 4x) - (4x^2 - 3x)(4x + 4)}{(2x^2 + 4x)^2}

12. Calculate the derivative of t(x) = (x^3 + x) / (2x^3 - 3x).
t'(x) = \frac{3x^2(2x^3 - 3x) - (x^3 + x)(6x^2 - 3)}{(2x^3 - 3x)^2}

13. Find the derivative of the function u(x) = (3x^2 + 4x) / (x^2 + x).
u'(x) = \frac{6x(x^2 + x) - (3x^2 + 4x)(2x + 1)}{(x^2 + x)^2}

14. Determine the derivative of v(x) = (2x^3 - x) / (4x^2 + 3x).
v'(x) = \frac{6x^2(4x^2 + 3x) - (2x^3 - x)(8x + 3)}{(4x^2 + 3x)^2}

15. Calculate the derivative of w(x) = (x^3 - x) / (3x^2 + 2x).
w'(x) = \frac{3x^2(3x^2 + 2x) - (x^3 - x)(6x + 2)}{(3x^2 + 2x)^2}