Calculate the derivative of the following functions.
y = sin (4x³ + 3x +1)

Verified step by step guidance

1. Step 1: Identify the outer function and the inner function. Here, the outer function is \( \sin(u) \) and the inner function is \( u = 4x^3 + 3x + 1 \).
2. Step 2: Apply the chain rule for differentiation, which states that \( \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \).
3. Step 3: Differentiate the outer function \( \sin(u) \) with respect to \( u \). The derivative is \( \cos(u) \).
4. Step 4: Differentiate the inner function \( u = 4x^3 + 3x + 1 \) with respect to \( x \). The derivative is \( 12x^2 + 3 \).
5. Step 5: Combine the results from steps 3 and 4 using the chain rule: \( \frac{dy}{dx} = \cos(4x^3 + 3x + 1) \cdot (12x^2 + 3) \).