Problem 1 - Calculus - AP Edition

Question

Find \(f^{\prime}(x)\). $$f(x)=4 \cos x+2 \sin x$$

Step-by-Step Solution

Answer

The derivative is \(f^{\prime}(x) = -4\sin x + 2\cos x\).

Step 1: Identify Differentiation Rules

To find the derivative of the function \(f(x) = 4\cos x + 2\sin x\), you need to use the differentiation rules for trigonometric functions. Recall that the derivative of \(\cos x\) is \(-\sin x\) and the derivative of \(\sin x\) is \(\cos x\).

Step 2: Differentiate Each Term Separately

Differentiate \(4\cos x\) and \(2\sin x\) separately. The derivative of \(4\cos x\) using the chain rule is \(4(-\sin x) = -4\sin x\). Similarly, the derivative of \(2\sin x\) is \(2(\cos x) = 2\cos x\).

Step 3: Combine the Derivatives

Combine the derivatives of the separate terms to find \(f^{\prime}(x)\). Thus, \(f^{\prime}(x) = -4\sin x + 2\cos x\).

Key Concepts

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions is crucial in calculus. Trigonometric functions, like \( \sin x \) and \( \cos x \), have specific rules for differentiation:

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

These derivatives can directly influence how we solve calculus problems related to periodic phenomena, like waves or circles.
For instance, in the given function \( f(x) = 4\cos x + 2\sin x \), we're essentially applying these simple differentiation rules to each trigonometric term.
The importance of getting comfortable with these trigonometric derivatives is expanding your capability to navigate complex calculus problems with ease.

Differentiation Rules

Differentiation rules make calculus problems manageable by offering clear strategies on how to approach the derivative of functions. When dealing with multiple terms, like \( f(x) = 4\cos x + 2\sin x \), it's essential to treat each term individually.
You'll often use the constant multiple rule which states that the derivative of \( c \cdot f(x) \) is \( c \cdot f^{\prime}(x) \), where \( c \) is a constant.

• The derivative of \( 4\cos x \) becomes \( 4(-\sin x) \) because of the constant multiple rule and the trigonometric derivative rule.
• Similarly, \( 2\sin x \) differentiates to \( 2\cos x \).

Remember, combining these rules effectively allows you to dissect more complex expressions into simpler bits, making the calculus process less daunting.

Chain Rule

In calculus, the chain rule is fundamental when dealing with composite functions, although it's not explicitly required in this example.
However, recognizing when to apply the chain rule is essential for other, more complex scenarios where a function is composed of another function. The chain rule is given by:\[\left( f(g(x)) \right)^{\prime} = f^\prime(g(x)) \cdot g^\prime(x)\]
In simpler terms, not only do you need to identify the inner and outer functions involved, but also differentiate each part correctly.
While our function here, \( 4\cos x + 2\sin x \), didn't require the use of the chain rule, familiarity with it is crucial as it frequently appears in differentiating nested expressions across calculus.