Find the first and second derivatives of the function.

*f*(*x*) = 

*x*2

7 + 4*x*

Given the function:

\[
f(x) = \frac{x^2}{7} + 4x
\]

Let's find its first and second derivatives.

### Step 1: First Derivative

The first derivative of \( f(x) \) with respect to \( x \) is denoted by \( f'(x) \). Differentiate each term separately:

- The derivative of \( \frac{x^2}{7} \) is \( \frac{2x}{7} \) (using the power rule).
- The derivative of \( 4x \) is \( 4 \).

Thus, the first derivative is:

\[
f'(x) = \frac{2x}{7} + 4
\]

### Step 2: Second Derivative

Now, we differentiate \( f'(x) \) to find the second derivative, denoted as \( f''(x) \):

- The derivative of \( \frac{2x}{7} \) is \( \frac{2}{7} \).
- The derivative of \( 4 \) is \( 0 \).

So the second derivative is:

\[
f''(x) = \frac{2}{7}
\]

### Final Answer:
- First derivative: \( f'(x) = \frac{2x}{7} + 4 \)
- Second derivative: \( f''(x) = \frac{2}{7} \)

Would you like any further details or have any questions?

### Related Questions:
1. How does the power rule work in differentiation?
2. What is the geometric interpretation of the first derivative?
3. How is the second derivative related to concavity of a function?
4. Can you find higher-order derivatives of this function?
5. How would the derivatives change if there were additional terms?

### Tip:
Always simplify terms as much as possible before differentiating—it can make the process quicker and reduce errors.

Find the first and second derivatives of the function. f(x) = x^2 / 7 + 4x

Explore the step-by-step process of finding the first and second derivatives of the function f(x) = x^2/7 + 4x. Learn the application of differentiation rules, including the Power Rule. Ideal for high school and college-level calculus students.

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