Problem 10 - Applied Calculus

Question

Differentiate the functions in Problems 1-20. Assume that \(A\) and \(B\) are constants. $$ y=2 \cos (5 t) $$

Step-by-Step Solution

Answer

The derivative is \(-10 \sin(5t)\).

Step 1: Identify the Function and Variables

The given function to differentiate is \( y = 2 \cos(5t) \). Here, \( t \) is the variable, and the rest are constants.

Step 2: Apply the Constant Multiple Rule

According to the Constant Multiple Rule, the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. This means we have: \( \frac{d}{dt}[2 \cos(5t)] = 2 \frac{d}{dt}[\cos(5t)] \).

Step 3: Use the Chain Rule

To differentiate \( \cos(5t) \), we need to apply the Chain Rule. The derivative of \( \cos(u) \), where \( u = 5t \), is \( -\sin(u) \cdot u' \). In this case, \( u' = 5 \). Thus, \( \frac{d}{dt}[\cos(5t)] = -\sin(5t) \cdot 5 \).

Step 4: Combine the Results

Substitute the result from Step 3 back into the expression from Step 2: \( 2 \cdot (-\sin(5t) \cdot 5) = -10 \sin(5t) \).

Step 5: Write the Final Derivative

The final derivative of the function \( y = 2 \cos(5t) \) with respect to \( t \) is \( \frac{dy}{dt} = -10 \sin(5t) \).

Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus used to differentiate composite functions. In simple terms, when you have one function nested inside another, you use the Chain Rule to take the derivative.

To apply the Chain Rule, you:

• Differentiate the outer function with respect to the inner function. This means you temporarily treat the inner part like a simple variable.
• Next, multiply that result by the derivative of the inner function itself.

For example, if you have a function like \( y = \cos(5t) \), you first differentiate \( \cos(u) \) with respect to \( u \) (where \( u = 5t \)). This derivative is \( -\sin(u) \). Then, multiply by the derivative of \( u \), which is \( 5 \). This gives:\[\frac{d}{dt}[\cos(5t)] = -\sin(5t) \cdot 5\]Using the Chain Rule, that's how you handle composite functions like \( \cos(5t) \) effectively.

Constant Multiple Rule

The Constant Multiple Rule is one of the most straightforward differentiation rules. It simplifies the process when a constant is multiplied by a function.

Here's how it works:

• Take the constant and move it outside the derivative operation.
• Differentiate the function as you normally would.
• Finally, multiply the derivative by the constant.

For the function \( y = 2 \cos(5t) \), the constant multiple is \( 2 \). By the rule, we calculate\[ \frac{d}{dt}[2 \cos(5t)] = 2 \cdot \frac{d}{dt}[\cos(5t)] \]This simplifies the calculation by focusing only on differentiating \( \cos(5t) \), and later adjusting by multiplying with the constant \( 2 \). This rule is particularly helpful when constants make functions seem complex.

Trigonometric Functions

Trigonometric functions often pop up in calculus, especially in physics and engineering problems. Understanding their derivatives is key to solving many real-world situations.

The primary trigonometric functions are sine \( \sin(x) \), cosine \( \cos(x) \), and tangent \( \tan(x) \). Each has a specific derivative:

• The derivative of \( \sin(x) \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \( -\sin(x) \).
• The derivative of \( \tan(x) \) is \( \sec^2(x) \).

In our problem, we are working with the cosine function. Knowing that \( \frac{d}{dt}[\cos(x)] = -\sin(x) \) helps us find that the derivative of \( \cos(5t) \) is \( -\sin(5t) \cdot 5 \).

Mastering these basic derivatives allows you to tackle more complex differentiation problems involving trigonometric functions.