It seems you've uploaded a document containing a few exercises related to differentiation, convexity, and variation. I'll break them down for you step by step.

Exercise 4: Derive $f(x) = \ln\left(\frac{2x - 1}{x + 2}\right) + e^{5x-1} \times (7x + 2)^3$

Let's differentiate this expression. First, we apply the derivative rules to each part:

1. The derivative of $\ln\left(\frac{2x - 1}{x + 2}\right)$ requires the chain rule.
2. The derivative of $e^{5x - 1} \times (7x + 2)^3$ will require both the product rule and the chain rule.

I will work through the derivation and provide the result for you shortly.

Exercise 5: Study the convexity of $f(x) = \ln(x^2 + 4x - 1)$

For convexity, we need to compute the second derivative of $f(x)$ and analyze its sign:

1. First, find the first derivative of $f(x)$.
2. Then, differentiate again to get the second derivative.
3. Analyze the sign of the second derivative to determine where the function is concave up or concave down.

Exercise 6: Derive $f(x) = \ln\left(\frac{x^3}{3} + \frac{x^2}{2} + x + 1\right)$

1. Derive $f(x)$: The derivative of this function involves applying the chain rule to the logarithmic function.
2. Show that $\frac{x^3}{3} + \frac{x^2}{2} + x + 1$ has a unique real root in the interval $[-2, -1]$: You will likely need to check for monotonicity or use intermediate value theorem or other methods to show the existence of a unique root.
3. Find the variation table of $f(x)$: This will involve computing the critical points and determining the increasing or decreasing intervals, then summarizing this information in a variation table.

Would you like me to go through each of these problems in more detail, or do you need assistance with a specific one? Feel free to ask!