91Ó°ÊÓ

The Product Rule is a fundamental concept in calculus, allowing us to find the derivative of a product of two functions. The rule states that if you have two differentiable functions, say \(f(\theta)\) and \(g(\theta)\), then the derivative of their product \(f(\theta)g(\theta)\) is:

• \( (f \cdot g)' = f' \cdot g + f \cdot g' \)

In our example, we apply the Product Rule to \(y(\theta) = \cos \theta \sin \theta\). First, we identify the components:

• \( f(\theta) = \cos \theta\) with derivative \(f'(\theta) = -\sin \theta\)
• \( g(\theta) = \sin \theta\) with derivative \(g'(\theta) = \cos \theta\)

This transforms the differentiation into an easier calculation:

• \(y'(\theta) = (-\sin \theta)(\sin \theta) + (\cos \theta)(\cos \theta)\)

This results in a simplified expression \(-\sin^2 \theta + \cos^2 \theta\), paving the way to compute higher-order derivatives.

Trigonometric functions, such as the sine and cosine functions, have derivatives that are well-defined and frequently used in calculus. Understanding these derivatives is key to handling more complex derivatives involving trigonometric expressions. In essence:

• The derivative of \( \sin \theta \) is \( \cos \theta \).
• The derivative of \( \cos \theta \) is \( -\sin \theta \).

These basic derivatives are integral to the solution of our function \( y(\theta) = \cos \theta \sin \theta \). When we found \( y'(\theta) = -\sin^2 \theta + \cos^2 \theta \), these rules allowed us to precisely calculate the necessary derivatives of sine and cosine without confusion. Recognizing these patterns helps in simplifying expressions and preparing for higher derivatives in calculus.

The Sum Rule for Derivatives is a straightforward yet powerful tool in calculus. It states that the derivative of a sum of functions is the sum of the derivatives of each individual function. Mathematically, if you have two functions \(h(\theta)\) and \(i(\theta)\), the derivative of their sum \(h(\theta) + i(\theta)\) is:

• \( (h + i)' = h' + i' \)

In our original problem, once we determined that the first derivative was \( y'(\theta) = -\sin^2 \theta + \cos^2 \theta \), we moved to find the second derivative. We redefined:

• \( h(\theta) = -\sin^2 \theta \) with derivative \( h'(\theta) = -2 \sin \theta \cos \theta\)
• \( i(\theta) = \cos^2 \theta \) with derivative \( i'(\theta) = -2 \sin \theta \cos \theta\)

By applying the Sum Rule, we calculated:

• \( y''(\theta) = h'(\theta) + i'(\theta) = -4 \sin \theta \cos \theta \)

The Sum Rule simplifies the process by breaking it down into smaller, manageable steps. This approach reveals that the second derivative function's behavior directly follows from each component function's derivative.