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

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.

Key Concepts

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if a function y is composed of two functions u and x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential for calculating derivatives of functions that are nested within one another.

Second Derivative

The second derivative of a function is the derivative of the derivative, providing information about the curvature of the function's graph. It indicates how the rate of change of the function is itself changing. In practical terms, the second derivative can reveal whether a function is concave up or concave down, which is useful for understanding the behavior of the function and identifying points of inflection.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in calculus, especially when dealing with periodic phenomena. The sine function, in particular, is crucial when applying the Chain Rule, as it often appears in composite functions. Understanding the properties and derivatives of trigonometric functions is essential for solving problems involving their rates of change and for applying rules like the Chain Rule effectively.