The Chain Rule for second derivatives
b. Use the formula in part (a) to calculate $\frac{d^2}{d{x}^2}(\sin (3x^4+5x^2+2))$.

Verified step by step guidance

1. Step 1: Identify the function and its inner function. Here, the outer function is \( \sin(u) \) and the inner function is \( u = 3x^4 + 5x^2 + 2 \).
2. Step 2: Compute the first derivative using the chain rule. The derivative of \( \sin(u) \) with respect to \( u \) is \( \cos(u) \), and the derivative of \( u \) with respect to \( x \) is \( 12x^3 + 10x \). Therefore, the first derivative is \( \frac{d}{dx} \left( \sin(u) \right) = \cos(u) \cdot (12x^3 + 10x) \).
3. Step 3: Apply the product rule to differentiate the first derivative. The first derivative is a product of two functions: \( \cos(u) \) and \( 12x^3 + 10x \). Use the product rule: \( (fg)' = f'g + fg' \).
4. Step 4: Differentiate \( \cos(u) \) with respect to \( x \) using the chain rule. The derivative of \( \cos(u) \) is \( -\sin(u) \cdot (12x^3 + 10x) \).
5. Step 5: Differentiate \( 12x^3 + 10x \) with respect to \( x \), which is \( 36x^2 + 10 \). Combine these results using the product rule to find the second derivative.