Problem 20 - Calculus of a Single Variable

Question

In Exercises 3–24, use the rules of differentiation to find the derivative of the function. $$ g(t)=\pi \cos t $$

Step-by-Step Solution

Answer

The derivative of the function \( g(t) = \pi \cos(t) \) is \( g'(t) = - \pi \sin(t) \)

Step 1: Identify the given function

The given function is \( g(t) = \pi \cos(t) \)

Step 2: Apply the derivative of cosine

The derivative of \( cos(t) \) is \(- \sin(t) \). Thus, by applying the chain rule, we get the derivative of our function: \( g'(t) = \pi \, \text{d/dt} [\cos(t)] = \pi (- \sin(t)) \)

Step 3: Simplify the derivative

Having applied the derivative rule, we simplify to get the final derivative of the function as: \( g'(t) = - \pi \sin(t) \)

Key Concepts

Derivative of Trigonometric Functions

When it comes to understanding how to differentiate trigonometric functions, the key is knowing the fundamental derivatives involved. For instance, the derivative of the cosine function, denoted as \( \cos(t) \), is \(-\sin(t)\). This means that when you take the derivative of \( \cos(t) \), the resulting function will have a negative sine, \(-\sin(t)\), as its result.

Trigonometric functions are periodic, meaning they repeat values in a regular interval. Therefore, their derivatives will also reflect this periodicity.

• For \(\sin(t)\), its derivative is \(\cos(t)\).
• The derivative of \(\tan(t)\) is \(\sec^2(t)\).
• Other functions like \(\cot(t), \sec(t), \text{and } \csc(t)\) have derivatives that follow their own specific patterns.

Understanding these basic derivatives forms the backbone for solving more complex calculus problems.

Chain Rule

The chain rule is a powerful tool in calculus, enabling us to differentiate compositions of functions. Simply put, it tells us how to find the derivative of a composite function. If you have a function \( f(g(x)) \), the derivative is found using the formula: \( (f(g(x)))' = f'(g(x)) \cdot g'(x) \).

This rule is particularly useful when dealing with functions like our trigonometric example, where a constant is multiplied with a trigonometric function. In our original exercise, the function \( g(t) = \pi \cos(t) \) was differentiated by recognizing \(\pi\) as a constant and applying the derivative \(-\sin(t)\) for \(\cos(t)\).

The key steps involved include:

• Identifying the outer and inner functions in your composition.
• Finding the derivative of both functions separately.
• Multiplying these derivatives together to get the result.

The chain rule often simplifies the process significantly, especially for complex or nested functions.

Calculus Exercises

Working on calculus exercises is an excellent way to internalize differentiation, especially with trigonometric functions and the chain rule. By practicing, you will become more familiar with the patterns and techniques used in calculus.

Considerations when tackling calculus problems include:

• Start by clearly identifying the type of functions involved and whether they're composite or simple.
• Review derivative rules for trigonometric functions to ensure you apply them correctly.
• Always simplify your result to its most straightforward form.

Simplifying the function makes sure your answer is not only correct but also easy to understand. Practice with a variety of problems to cover different function scenarios. The more you practice, the more intuitive these calculations will become, helping you draw parallels and see patterns across different problems.