Question

Compute the derivative of the given function. $$f(x)=\sin (3 x+4) \cos (5-2 x)$$

Short answer

The function given is a product of two functions: \( u = \sin (3x + 4) \) and \( v = \cos (5 - 2x) \). According to the product rule, the derivative \( (uv)' \) is given by \( u'v + uv' \).

Step-by-step solution

Identify the Product Rule

The function given is a product of two functions: \( u = \sin (3x + 4) \) and \( v = \cos (5 - 2x) \). According to the product rule, the derivative \( (uv)' \) is given by \( u'v + uv' \).

Key concepts

Product Rule

In calculus, when you need to find the derivative of a function that is a product of two other functions, the product rule comes into play. It's a vital rule that helps differentiate products without having to expand or simplify. The product rule states:

• If you have two functions, let's call them \( u(x) \) and \( v(x) \), their product \( y = u(x) \cdot v(x) \) has a derivative given by the formula: \( y' = u'(x) \cdot v(x) + u(x) \cdot v'(x) \).
• This means you first derive \( u(x) \) while keeping \( v(x) \) intact. Then derive \( v(x) \) while keeping \( u(x) \) intact, and add the results together.

For the original exercise, the given function \( f(x) = \sin(3x+4) \cdot \cos(5-2x) \) needs you to apply the product rule.
You identify the two component functions as:

• \( u = \sin(3x+4) \)
• \( v = \cos(5-2x) \)

Each function has to be independently differentiated and combined as per the product rule formula.

Trigonometric Functions

Trigonometric functions like sine and cosine are foundational in calculus, particularly in differentiation. These functions oscillate, meaning they go up and down in a cyclical manner, which influences their derivatives.
For sine and cosine functions, their derivatives have specific patterns:

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

In more complex expressions, these rules allow you to handle derivatives confidently.
With the function \( u = \sin(3x+4) \), you apply the chain rule. Chain rule is used since the argument of the sine function is more than just \( x \). Similarly, for \( v = \cos(5-2x) \), the derivative involves the chain rule due to the argument \( 5-2x \).
The chain rule lets you differentiate composite functions by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

Derivatives

Derivatives represent the rate of change of functions. In simpler terms, they show how a function's output value changes in response to small changes in input values.
In calculus, especially when dealing with functions like polynomials, trigonometric, exponential, etc., derivatives provide a way to measure instantaneous change and are fundamental tools.
When finding the derivative of a product of trigonometric functions, you combine different rules like the product rule and chain rule to achieve the final derivative. This enables a step by step approach:

• Find the derivative of each component function (use the chain rule for trigonometric functions with complex arguments).
• Apply the product rule using the derived components.
• Simplify the expression if possible.

Through this process, you can systematically tackle complex differentiation problems and gain insights into the behavior of different types of functions under change.