whats the first derivitive of y=sin(3x^2+cosx)

The first derivative of the function \( y = \sin(3x^2 + \cos x) \) can be found using the chain rule.

### Step-by-step solution:
1. Let \( u = 3x^2 + \cos x \), then \( y = \sin(u) \).
2. The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \), so:
   \[
   \frac{dy}{du} = \cos(u)
   \]
3. Now, apply the chain rule. The derivative of \( y \) with respect to \( x \) is:
   \[
   \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
   \]
4. First, differentiate \( u = 3x^2 + \cos x \) with respect to \( x \):
   \[
   \frac{du}{dx} = 6x - \sin x
   \]
5. Now, substitute \( u \) and \( \frac{du}{dx} \) into the expression for \( \frac{dy}{dx} \):
   \[
   \frac{dy}{dx} = \cos(3x^2 + \cos x) \cdot (6x - \sin x)
   \]

Thus, the first derivative of \( y = \sin(3x^2 + \cos x) \) is:
\[
\frac{dy}{dx} = \cos(3x^2 + \cos x) \cdot (6x - \sin x)
\]

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### Related questions:
1. What is the second derivative of \( y = \sin(3x^2 + \cos x) \)?
2. How does the chain rule apply to more complex functions?
3. Can you explain the product rule in differentiation?
4. How do you differentiate trigonometric functions in general?
5. What are some real-world applications of derivatives?

**Tip:** When differentiating composite functions, always identify the inner and outer functions to apply the chain rule correctly.

whats the first derivative of y=sin(3x^2+cosx)

Find the first derivative of the function y = sin(3x^2 + cos(x)) using the chain rule. This problem explores key calculus concepts like differentiation, chain rule, and trigonometric functions. Suitable for students in Grades 11-12 or college-level calculus.

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