Problem 9 - Applied Calculus

Question

Differentiate each function. $$ f(t)=2 \cos 3 t $$

Step-by-Step Solution

Answer

The derivative is \(-6\sin(3t)\).

Step 1: Identify the function structure

The function given is a trigonometric function that can be represented as \( f(t) = 2 \cdot \cos(3t) \). It involves a constant multiple and a composed function pattern.

Step 2: Apply the derivative rule for cosine

The derivative of \( \cos(u) \) is \( -\sin(u) \) where \( u \) is a function of \( t \). For our function \( 2\cos(3t) \), we have \( u = 3t \). Thus, we apply this rule to find the derivative of the cosine part.

Step 3: Use the chain rule

Since \( u = 3t \), apply the chain rule. The chain rule states that the derivative of \( \cos(3t) \) becomes \( -\sin(3t) \cdot (d/dt)(3t) \). Calculate \( (d/dt)(3t) = 3 \).

Step 4: Differentiate the entire function

Combine the chain rule result with the constant multiple: Apply \( (d/dt)[2 \cdot \cos(3t)] = 2 \cdot [-\sin(3t) \cdot 3] \). This results in \( -6\sin(3t) \).

Step 5: Verify and simplify if necessary

Check the simplified expression for accuracy. The differentiation has resulted in the expression \( -6\sin(3t) \), which is accurate according to derivative rules for trigonometric functions.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, play an important role in calculus. They appear frequently and have their unique properties when differentiated.

• Cosine Function: The function \( \cos(u) \) returns the cosine of an angle \( u \), often found in terms related to waves and circular motion.
• Understanding the Context: In our exercise, \( f(t) = 2 \cos(3t) \), the function represents a scaled and compressed cosine wave.

Trigonometric functions like cosine are essential for modeling periodic phenomena. Differentiation of these functions helps to understand how rate and behavior change over time.

Derivative Rules

Derivative rules offer essential tools in calculus for finding the rate of change of functions. The key to mastering derivation is knowing the basic rules and how they apply to different types of functions, including trigonometric ones.

• Cosine Derivative Rule: The derivative of \( \cos(u) \) is found using the rule that it becomes \( -\sin(u) \).
• Constant Multiple Rule: If you are taking the derivative of a function multiplied by a constant, like \( 2 \cdot \cos(3t) \), you can differentiate the function normally and then multiply the result by the constant.

In our problem, these rules let us transform \( f(t) = 2 \cos(3t) \) into its derivative \( -6 \sin(3t) \), reflecting the rate of change of our initial cosine function.

Chain Rule

The chain rule is a crucial differentiation tool used for functions composed of other functions, commonly known as composite functions. It simplifies the process of taking derivatives of these more complex structures.

• Purpose: The chain rule allows us to differentiate a function within a function efficiently.
• Formula: In simple terms, for a composite function \( f(g(x)) \), its derivative is \( f'(g(x)) \cdot g'(x) \).

In our exercise, the chain rule is applied to \( \cos(3t) \), as the inner function \( g(t) = 3t \) changes the argument of the cosine function. By applying the chain rule: differentiate \( \cos(3t) \) as \( -\sin(3t) \), then multiply by the derivative of \( 3t \), which is \( 3 \), yielding the result \( -6\sin(3t) \). This results in the precise rate of change for this composite function.