91Ó°ÊÓ

In calculus, when you need to find the derivative of a product of two functions, the product rule is an essential tool. The product rule helps you differentiate expressions where two functions are multiplied together. For the function given in your exercise, you can see that it consists of two functions:

• \( u = \sin(\theta^2) \)
• \( v = \cos(2\theta) \)

The product rule formula is \((uv)' = u'v + uv'\), which means:

• First, differentiate the first function \( u \), while leaving the second function \( v \) as it is.
• Then, add the product of the first function \( u \) and the derivative of the second function \( v \).

This method ensures that you account for the change in both components of the product as they change with \( \theta \). Understanding this rule is critical, particularly when dealing with complex expressions that are products of simpler functions.

The chain rule is another vital technique used for finding derivatives, especially when dealing with composite functions. A composite function is simply a function inside another function, like \( \sin(\theta^2) \) or \( \cos(2\theta) \).
When applying the chain rule, you differentiate the outer function first and then multiply it by the derivative of the inner function.

• For example, for \( u = \sin(\theta^2) \), treat \( \sin(x) \) as the outer function and \( \theta^2 \) as the inner function. The derivative becomes \( \cos(\theta^2) \cdot 2\theta \).
• Similarly, for \( v = \cos(2\theta) \), \( \cos(x) \) is the outer function and \( 2\theta \) is the inner function. The derivative results in \(-2\sin(2\theta) \).

This approach efficiently handles derivatives of more complex functions by peeling off layers and addressing each separately in a sequential manner.

Trigonometric functions like sine and cosine have specific derivatives that you will frequently use in calculus. The equations for these derivatives are fundamental building blocks in solving more complex problems. Understanding these helps make the process smoother.

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

In this problem, for \( \sin(\theta^2) \) and \( \cos(2\theta) \), use these basic derivatives in combination with the chain rule. Notice that trigonometric derivatives maintain their basic form; you just apply the chain rule to account for the whole expression.
So, whenever you see "\( \sin \)" or "\( \cos \)" in a function, remember these rules. They will come in handy when you calculate the derivatives involving trigonometric parts of any expression.