The definition of the derivative is given by:

$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

We will compute $f'(x)$ for each function.

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(a) $f(x) = 4x + 2$

$f(x+h) = 4(x+h) + 2 = 4x + 4h + 2$

$\frac{f(x+h) - f(x)}{h} = \frac{(4x + 4h + 2) - (4x + 2)}{h} = \frac{4h}{h} = 4$

$\lim_{h \to 0} 4 = 4$

Answer: $f'(x) = 4$

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(b) $f(x) = 3x^2 + 2x + 5$

$f(x+h) = 3(x+h)^2 + 2(x+h) + 5$

Expanding: $f(x+h) = 3(x^2 + 2xh + h^2) + 2x + 2h + 5$

$= 3x^2 + 6xh + 3h^2 + 2x + 2h + 5$

$\frac{f(x+h) - f(x)}{h} = \frac{(3x^2 + 6xh + 3h^2 + 2x + 2h + 5) - (3x^2 + 2x + 5)}{h}$

$= \frac{6xh + 3h^2 + 2h}{h} = 6x + 3h + 2$

$\lim_{h \to 0} (6x + 3h + 2) = 6x + 2$

Answer: $f'(x) = 6x + 2$

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(c) $f(x) = \sqrt{x+3}$ at $x = -2$

$f(x+h) = \sqrt{x+h+3}$

$\frac{f(x+h) - f(x)}{h} = \frac{\sqrt{x+h+3} - \sqrt{x+3}}{h}$

Multiply numerator and denominator by the conjugate:

$\frac{(\sqrt{x+h+3} - \sqrt{x+3})(\sqrt{x+h+3} + \sqrt{x+3})}{h (\sqrt{x+h+3} + \sqrt{x+3})}$

$= \frac{(x+h+3) - (x+3)}{h (\sqrt{x+h+3} + \sqrt{x+3})} = \frac{h}{h (\sqrt{x+h+3} + \sqrt{x+3})}$

$= \frac{1}{\sqrt{x+h+3} + \sqrt{x+3}}$

Taking the limit as $h \to 0$:

$f'(-2) = \frac{1}{2\sqrt{-2+3}} = \frac{1}{2\sqrt{1}} = \frac{1}{2}$

Answer: $f'(-2) = \frac{1}{2}$

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(d) $f(x) = \frac{1}{x}$

$f(x+h) = \frac{1}{x+h}$

$\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{x+h} - \frac{1}{x}}{h}$

Finding a common denominator:

$\frac{\frac{x - (x+h)}{x(x+h)}}{h} = \frac{\frac{-h}{x(x+h)}}{h} = \frac{-1}{x(x+h)}$

$\lim_{h \to 0} \frac{-1}{x(x+h)} = \frac{-1}{x^2}$

Answer: $f'(x) = -\frac{1}{x^2}$

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(e) $f(x) = \frac{x}{x+4}$

Using the quotient rule:

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

$= \frac{x+4-x}{(x+4)^2} = \frac{4}{(x+4)^2}$

Answer: $f'(x) = \frac{4}{(x+4)^2}$

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Do you want more details on any step?

Here are five related questions you might find helpful:

1. How do you find the derivative using the limit definition for trigonometric functions?
2. Can you explain the quotient rule in more detail with another example?
3. What is the difference between derivative rules and the limit definition?
4. How does the definition of the derivative relate to the concept of instantaneous rate of change?
5. What are common mistakes students make when computing derivatives using the limit definition?

Tip: Always simplify algebraic expressions before taking the limit to make calculations easier! 🚀