https://file.aiquickdraw.com/mathbot/file/17295486926051j2nmz91.jpg

The function given in the image is:

\[
f(x) = 3\cos(\sin x)
\]

To find the derivative, \( f'(x) \), we can apply the chain rule. The chain rule states that if a function is composed of other functions, the derivative is the derivative of the outer function multiplied by the derivative of the inner function.

1. The outer function is \( 3\cos(u) \), where \( u = \sin x \). The derivative of \( 3\cos(u) \) with respect to \( u \) is:

\[
\frac{d}{du}[3\cos(u)] = -3\sin(u)
\]

2. The inner function is \( u = \sin x \), and the derivative of \( \sin x \) with respect to \( x \) is:

\[
\frac{d}{dx}[\sin x] = \cos x
\]

Now, applying the chain rule:

\[
f'(x) = (-3\sin(\sin x)) \cdot \cos x
\]

Thus, the derivative is:

\[
f'(x) = -3\sin(\sin x)\cos x
\]

Let me know if you would like further clarification!

Here are some related questions:
1. How would the derivative change if the function were \( f(x) = 3\sin(\sin x) \)?
2. Can you apply the chain rule to a different trigonometric function, such as \( f(x) = 2\sin(\cos x) \)?
3. What happens to the derivative if the constant 3 is removed, i.e., \( f(x) = \cos(\sin x) \)?
4. How does the chain rule apply if the outer function is squared, e.g., \( f(x) = \cos^2(\sin x) \)?
5. What is the second derivative \( f''(x) \) of the given function?

**Tip:** When using the chain rule, always differentiate from the outermost function to the innermost, multiplying the derivatives of each layer.

Find the derivative of the function f(x) = 3cos(sin(x))

Learn how to find the derivative of the function f(x) = 3cos(sin(x)) by applying the Chain Rule. This problem involves differentiating trigonometric functions and is suitable for university-level calculus students.

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