Problem 52 - Calculus - AP Edition

Question

Find \(d^{2} y / d x^{2}\) $$y=\sin \left(3 x^{2}\right)$$

Step-by-Step Solution

Answer

The second derivative is \(\frac{d^2 y}{d x^2} = 6 \cos(3x^2) - 36x^2 \sin(3x^2)\).

Step 1: Identify the Outer Function

The function given is \(y = \sin(3x^2)\). We need to identify the outer function, which is \(\sin(u)\) in this case, where \(u = 3x^2\).

Step 2: Differentiate Using Chain Rule

Apply the chain rule. The derivative of \(y = \sin(u)\) is \(\cos(u)\). So, \(\frac{dy}{du} = \cos(3x^2)\). Also differentiate \(u = 3x^2\) with respect to \(x\), giving \( \frac{du}{dx} = 6x\). Thus, \(\frac{dy}{dx} = \cos(3x^2) \cdot 6x\).

Step 3: First Derivative Simplification

Combine the derivatives: \(\frac{dy}{dx} = 6x \cos(3x^2)\).

Step 4: Differentiate to Find the Second Derivative

To find the second derivative, we need to differentiate \(\frac{dy}{dx} = 6x \cos(3x^2)\) with respect to \(x\). Apply the product rule for \(uv\), where \(u = 6x\) and \(v = \cos(3x^2)\).

Step 5: Apply the Product Rule

The product rule states \(\frac{d}{dx} (uv) = u'v + uv'\). Here \( u' = 6\) and \(v = \cos(3x^2)\), so \(v' = -\sin(3x^2) \cdot (6x)\) using the chain rule. Thus, \( \frac{d}{dx}(6x \cos(3x^2)) = 6 \cos(3x^2) - 36x^2 \sin(3x^2) \).

Step 6: Simplify the Second Derivative

Combine the terms: \(\frac{d^2y}{dx^2} = 6 \cos(3x^2) - 36x^2 \sin(3x^2)\).

Key Concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus used when dealing with composite functions. A composite function is created when one function is applied within another. In this case, we have the function \( y = \sin(3x^2) \). Here, \( \sin(u) \) is the outer function, and \( u = 3x^2 \) is the inner function. The purpose of the Chain Rule is to differentiate the outer function with respect to the inner function and then multiply by the derivative of the inner function with respect to \( x \).

To apply the Chain Rule:

• Find the derivative of the outer function. Here, the derivative of \( \sin(u) \) is \( \cos(u) \).
• Differentiate the inner function. For \( u = 3x^2 \), the derivative is \( 6x \).
• Multiply both results. Thus, \( \frac{dy}{dx} = \cos(3x^2) \cdot 6x \).

Understanding and applying the Chain Rule correctly is crucial in obtaining accurate derivatives, especially when dealing with functions wrapped inside other functions.

Product Rule

The Product Rule is used when we need to take the derivative of a product of two functions. It's very handy when you have expressions like \( 6x \cos(3x^2) \), which is what we encounter when working on the second derivative of our original function.

The Product Rule states that if we have a function \( y = uv \), where both \( u \) and \( v \) are functions of \( x \), then the derivative \( \frac{d}{dx}(uv) \) is \( u'v + uv' \). Consider:

• Let \( u = 6x \) and \( v = \cos(3x^2) \).
• Differentiate each part: \( u' = 6 \) and for \( v' \), apply the chain rule to get \( v' = -\sin(3x^2) \cdot 6x \).
• Plug these into the formula to get \( 6 \cos(3x^2) - 36x^2 \sin(3x^2) \).

In math problems that mix multiple function types, knowing and practicing the Product Rule helps tackle challenging differentiation tasks successfully.

Trigonometric Functions

Trigonometric functions are an essential part of calculus, and they often appear in various forms in different problems. Functions like \( \sin \) and \( \cos \) represent simple harmonic motion and periodic behavior and are crucial in understanding the nature of waves and cycles in mathematics.

In the original exercise, \( \sin(3x^2) \) is the primary trigonometric function. When differentiating trigonometric functions like these, remember:

• For \( \sin(u) \), the derivative is \( \cos(u) \).
• For \( \cos(u) \), the derivative is \( -\sin(u) \).
• These derivatives allow us to unlock more complex interrelations in calculus, such as rates of change in harmonic motion.

Having a strong grasp of how to handle these derivatives empowers students to master dynamic systems and wave-like patterns in mathematical analysis better.