a. Find the derivative function f` for the function f.
spuareroot(3x+4); a=7 show steps

We are tasked with finding the derivative of the function \( f(x) = \sqrt{3x + 4} \) at \( a = 7 \). Here’s how we can do that, step by step:

### Step 1: Rewrite the function using exponents
We can rewrite the square root function as a fractional exponent for easier differentiation:

\[
f(x) = \sqrt{3x + 4} = (3x + 4)^{\frac{1}{2}}
\]

### Step 2: Apply the chain rule
To differentiate \( f(x) = (3x + 4)^{\frac{1}{2}} \), we use the **chain rule**. The chain rule states that if a function is composed of two functions, say \( f(g(x)) \), then the derivative is:

\[
f'(x) = \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)
\]

For \( f(x) = (3x + 4)^{\frac{1}{2}} \), let \( u = 3x + 4 \), then the function becomes \( f(x) = u^{\frac{1}{2}} \).

1. **Differentiate the outer function** \( u^{\frac{1}{2}} \):

\[
\frac{d}{du}(u^{\frac{1}{2}}) = \frac{1}{2} u^{-\frac{1}{2}} = \frac{1}{2} (3x + 4)^{-\frac{1}{2}}
\]

2. **Differentiate the inner function** \( u = 3x + 4 \):

\[
\frac{d}{dx}(3x + 4) = 3
\]

### Step 3: Combine using the chain rule
Now, multiply the derivatives from the chain rule:

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

This simplifies to:

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

### Step 4: Evaluate at \( x = 7 \)
Now, substitute \( x = 7 \) into the derivative:

\[
f'(7) = \frac{3}{2} (3(7) + 4)^{-\frac{1}{2}} = \frac{3}{2} (21 + 4)^{-\frac{1}{2}} = \frac{3}{2} (25)^{-\frac{1}{2}}
\]

Since \( 25^{-\frac{1}{2}} = \frac{1}{5} \), we have:

\[
f'(7) = \frac{3}{2} \cdot \frac{1}{5} = \frac{3}{10}
\]

### Final Answer:
The derivative of \( f(x) = \sqrt{3x + 4} \) is:

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

And at \( x = 7 \), the derivative is:

\[
f'(7) = \frac{3}{10}
\]

Would you like more details on any part of the process? Here are some related questions:

1. How does the chain rule apply to composite functions in general?
2. What is the significance of evaluating the derivative at \( x = 7 \)?
3. Can we use this method for more complex square root functions?
4. How can the power rule and chain rule be combined in higher-order derivatives?
5. What is the interpretation of the derivative \( f'(7) = \frac{3}{10} \) in real-world terms?

**Tip:** When differentiating square root functions, it's often easier to rewrite them with fractional exponents.

Find the derivative function f' for the function f: sqrt(3x + 4); a = 7 show steps

Find the derivative of the function f(x) = sqrt(3x + 4) using the chain rule and evaluate it at x = 7. Detailed steps include applying the chain rule and power rule, making this an essential problem for calculus students in Grades 11-12.

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