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.