Let's analyze the problem step by step based on the image provided.

We are given a function:

$f(x) = 10x(\sin(x) + \cos(x))$

Step 1: Finding $f'(x)$

We are already provided with the derivative $f'(x)$ in the image:

$f'(x) = 10 \sin(x) + 10 \cos(x) + 10x \cos(x) - 10x \sin(x)$

Step 2: Finding $f'\left( -\frac{\pi}{6} \right)$

Now, we need to substitute $x = -\frac{\pi}{6}$ into the derivative expression and simplify:

1. $\sin\left( -\frac{\pi}{6} \right) = -\frac{1}{2}$
2. $\cos\left( -\frac{\pi}{6} \right) = \frac{\sqrt{3}}{2}$

Using these values, we substitute $x = -\frac{\pi}{6}$ into the expression for $f'(x)$:

$f'\left( -\frac{\pi}{6} \right) = 10\left(-\frac{1}{2}\right) + 10\left(\frac{\sqrt{3}}{2}\right) + 10\left(-\frac{\pi}{6}\right)\left(\frac{\sqrt{3}}{2}\right) - 10\left(-\frac{\pi}{6}\right)\left(-\frac{1}{2}\right)$

Now, I'll calculate the exact value for this derivative at $x = -\frac{\pi}{6}$.The value of $f'\left( -\frac{\pi}{6} \right)$ is approximately $-3.49$.

Would you like further details on this calculation, or do you have any questions?

Here are 5 related questions for further exploration:

1. How would the result change if the function involved different trigonometric identities, like $\tan(x)$?
2. What are the rules for differentiating a product of two functions, like $x \sin(x)$?
3. How can the derivative of a function change the behavior of its graph?
4. What is the second derivative of $f(x) = 10x(\sin(x) + \cos(x))$, and what does it signify?
5. How does the chain rule apply when differentiating composite trigonometric functions?

Tip: Always check if the trigonometric values for standard angles like $\frac{\pi}{6}$ or $\frac{\pi}{4}$ are familiar, as they can simplify calculations significantly.