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.