To compute the derivative of the function $f(t) = \sqrt{3t + 1}$ using the definition of the derivative, we apply the limit definition of the derivative:

$f'(t) = \lim_{h \to 0} \frac{f(t+h) - f(t)}{h}$

Step-by-Step Derivation:

Given $f(t) = \sqrt{3t + 1}$, we need to compute:

$f'(t) = \lim_{h \to 0} \frac{\sqrt{3(t+h) + 1} - \sqrt{3t + 1}}{h}$

1. Simplify the expression:

   Start by expanding $3(t+h) + 1$ inside the square root:

   $f'(t) = \lim_{h \to 0} \frac{\sqrt{3t + 3h + 1} - \sqrt{3t + 1}}{h}$

2. Rationalize the numerator:

   To simplify the difference of square roots, multiply and divide by the conjugate:

   $f'(t) = \lim_{h \to 0} \frac{\left( \sqrt{3t + 3h + 1} - \sqrt{3t + 1} \right) \left( \sqrt{3t + 3h + 1} + \sqrt{3t + 1} \right)}{h \left( \sqrt{3t + 3h + 1} + \sqrt{3t + 1} \right)}$

   This simplifies the numerator to the difference of squares:

   $f'(t) = \lim_{h \to 0} \frac{(3t + 3h + 1) - (3t + 1)}{h \left( \sqrt{3t + 3h + 1} + \sqrt{3t + 1} \right)}$

   Simplify the terms:

   $f'(t) = \lim_{h \to 0} \frac{3h}{h \left( \sqrt{3t + 3h + 1} + \sqrt{3t + 1} \right)}$

3. Cancel $h$:

   $f'(t) = \lim_{h \to 0} \frac{3}{\sqrt{3t + 3h + 1} + \sqrt{3t + 1}}$

4. Evaluate the limit as $h \to 0$:

   As $h \to 0$, the expression simplifies to:

   $f'(t) = \frac{3}{2 \sqrt{3t + 1}}$

Final Answer:

$f'(t) = \frac{3}{2 \sqrt{3t + 1}}$

Would you like further explanation on any step, or is there another question related to derivatives that you'd like to explore? Here are some related questions:

1. What is the second derivative of $f(t) = \sqrt{3t + 1}$?
2. How do you apply the derivative formula for functions involving higher powers or products?
3. Can the chain rule be used to differentiate functions like $\sqrt{3t + 1}$?
4. How can you differentiate rational functions using limits?
5. How does the derivative of a square root function compare with the derivative of a power function?

Tip: When differentiating functions involving square roots, it's often useful to rationalize the numerator or use the chain rule for efficient computation!