Find the first and second derivatives of the function.

*f*(*x*) = 

*x*2

7 + 4*x*

The given function is:

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

### Step 1: First Derivative
We need to differentiate \( f(x) \) with respect to \( x \).

\[
f'(x) = \frac{d}{dx} \left( \frac{x^2}{7} \right) + \frac{d}{dx} \left( 4x \right)
\]

Using the power rule for differentiation (\( \frac{d}{dx}(x^n) = n x^{n-1} \)) and the fact that the derivative of a constant times a variable is the constant:

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

### Step 2: Second Derivative
Now, we differentiate \( f'(x) \) to find the second derivative.

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

Since the derivative of \( \frac{2x}{7} \) is \( \frac{2}{7} \) and the derivative of a constant is 0:

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

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

Would you like any further details on the differentiation process?

Here are 5 related questions:
1. How do you find the derivative of a polynomial function in general?
2. What is the power rule for differentiation and how is it applied?
3. What are the applications of first and second derivatives in real-life scenarios?
4. How do second derivatives help in understanding the concavity of a function?
5. Can you explain the relationship between the derivative and the slope of a tangent line?

**Tip:** The second derivative gives insight into the concavity of the function. If \( f''(x) > 0 \), the function is concave up, and if \( f''(x) < 0 \), it's concave down.

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

Learn how to find the first and second derivatives of the function f(x) = x^2 / 7 + 4x. This detailed solution applies the power rule and sum rule for differentiation. Suitable for students studying calculus at Grades 10-12.

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