Using definition, compute the derivative function of 𝑓𝑓 (𝑥𝑥) below:

1. 𝑓𝑓 (𝑥𝑥) =

2. 𝑓𝑓(𝑡𝑡) = √3𝑡𝑡 + 1

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!

Using definition, compute the derivative function of 𝑓(𝑡) = √(3𝑡 + 1).

Learn how to compute the derivative of the function f(t) = √(3t + 1) using the limit definition of the derivative. This solution involves simplifying expressions, rationalizing the numerator, and evaluating limits. Suitable for students in Grades 11-12 studying calculus.

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