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The problem provides the following function:

\[
f(x) = -2\cos(x) + 10\tan(x)
\]

### 1. Derivative of \( f(x) \)
The first derivative of \( f(x) \) is given as:

\[
f'(x) = 2\sin(x) + 10\sec^2(x)
\]

### 2. Evaluate \( f'( \frac{5\pi}{4} ) \)

We need to calculate \( f'\left(\frac{5\pi}{4}\right) \). Let's break this down:

1. \( \sin\left(\frac{5\pi}{4}\right) = \sin\left(\pi + \frac{\pi}{4}\right) = -\sin\left(\frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
2. \( \sec\left(\frac{5\pi}{4}\right) = \sec\left(\pi + \frac{\pi}{4}\right) = -\sec\left(\frac{\pi}{4}\right) = -\sqrt{2} \)

Now, substitute these values into the derivative:

\[
f'\left(\frac{5\pi}{4}\right) = 2\left(-\frac{\sqrt{2}}{2}\right) + 10(-\sqrt{2})^2
\]
\[
f'\left(\frac{5\pi}{4}\right) = -\sqrt{2} + 10(2)
\]
\[
f'\left(\frac{5\pi}{4}\right) = -\sqrt{2} + 20
\]

This is the simplified value of the derivative at \( \frac{5\pi}{4} \).

Would you like me to provide further details or evaluate the final expression? Also, here are some related questions for practice:

1. What is the derivative of \( \tan(x) \) and \( \sec(x) \)?
2. How does the behavior of trigonometric functions change in different quadrants?
3. What are the key identities for simplifying trigonometric expressions?
4. How would you apply the chain rule to more complex trigonometric functions?
5. Can you find the second derivative \( f''(x) \)?

### Tip:
When evaluating trigonometric functions at specific angles, use reference angles and identities to simplify your calculations.

Let f(x) = -2cos(x) + 10tan(x). 1. Find the derivative f'(x). 2. Evaluate f'(5π/4).

Learn how to find the derivative of f(x) = -2cos(x) + 10tan(x) and evaluate it at f'(5π/4). This problem involves key calculus and trigonometry concepts, including differentiation rules for trigonometric functions. Suitable for Grades 11-12.

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