To find the derivative of the function $f(x) = 7\sin(x) + 2\cos(x)$, we will apply the basic rules of differentiation for sine and cosine functions.

The derivatives of the basic trigonometric functions are:

• $\frac{d}{dx} \sin(x) = \cos(x)$
• $\frac{d}{dx} \cos(x) = -\sin(x)$

Now, differentiate each term in the function $f(x)$:

$\frac{d}{dx} [7\sin(x)] = 7\cos(x)$ $\frac{d}{dx} [2\cos(x)] = -2\sin(x)$

Thus, the derivative of $f(x)$ is:

$f'(x) = 7\cos(x) - 2\sin(x)$

Final Answer:

$f'(x) = 7\cos(x) - 2\sin(x)$

Do you need any further details or clarifications?

Here are five related questions to expand your understanding:

1. What is the second derivative of $f(x)$?
2. How would you find the critical points of $f(x)$?
3. Can you integrate $f(x) = 7\sin(x) + 2\cos(x)$?
4. What is the amplitude of the function $f(x)$?
5. How would you sketch the graph of $f(x)$?

Tip: When differentiating trigonometric functions, always remember the standard derivatives of sine and cosine.