https://file.aiquickdraw.com/mathbot/file/17258489709328ail5apl.jpg

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.

Let f(x) = 10x(sin(x) + cos(x)). Find the following: 1. f'(x) and 2. f'(-π/6)

Learn how to differentiate the function f(x) = 10x(sin(x) + cos(x)) using the product rule and evaluate the derivative at x = -π/6. Includes key trigonometric identities and step-by-step simplifications. Suitable for high school calculus students.

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