Problem 1 - Single Variable Calculus

Question

Find the derivative of the function. $$ f(x)=4 \cos x-2 x+1 $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x) = 4\cos x - 2x + 1\) is \(f'(x) = -4\sin x - 2\).

Step 1: Identify the individual terms of the function

The given function f(x) can be represented as a sum/difference of three separate terms: \(f(x) = \) (4 * cos(x)) - (2 * x) + 1

Step 2: Differentiate each term

According to the sum/difference rule, the derivative of a sum/difference of functions is the sum/difference of their derivatives. We will now differentiate each term: 1. Differentiate 4 * cos(x): Apply both the constant rule and the derivative of cos(x): \( \frac{d}{dx}(4\cos x)= 4\frac{d}{dx}\cos x \) The derivative of cos(x) is - sin(x): \( \frac{d}{dx}(4\cos x)= 4(-\sin x)= -4\sin x \) 2. Differentiate 2 * x: Apply the constant rule and the derivative of x: \( \frac{d}{dx}(2x)= 2\frac{d}{dx}(x) \) The derivative of x is 1: \( \frac{d}{dx}(2x)= 2(1)= 2 \) 3. Differentiate 1: Since 1 is a constant, its derivative will be 0: \( \frac{d}{dx}(1) = 0 \)

Step 3: Combine the derivatives to obtain the final solution

Now, we will put together the derivatives of each term to find the derivative of f(x): \(f'(x) = -4\sin x - 2 + 0 \) Simplifying the expression, we get: \(f'(x) = -4\sin x - 2 \) The derivative of the function f(x) = 4 * cos(x) - 2x + 1 is \(f'(x) = -4\sin x - 2\).

Key Concepts

Sum/Difference Rule

Derivative calculations often begin by breaking down complex functions into simpler parts. The sum/difference rule is particularly useful for this purpose. When you have a function formed by the sum or difference of several terms, such as \(f(x) = 4\cos x - 2x + 1\), you can differentiate each component separately.

Here's why this is beneficial:

• It simplifies complex expressions into manageable pieces.
• Allows for the use of specific derivative rules for each term.
• Ensures clarity and accuracy in calculating derivatives.

In our example, we separated the terms as \(4 \cos x\), \(-2x\), and \(1\). We then applied the sum/difference rule, which states: if \(f(x) = u(x) + v(x)\) or \(f(x) = u(x) - v(x)\), then \(f'(x) = u'(x) + v'(x)\) or \(f'(x) = u'(x) - v'(x)\). This approach makes the derivative much easier to compute.

Constant Rule

The constant rule is a straightforward yet essential tool in calculus. It helps when differentiating terms that involve constants. Essentially, if you have a constant multiplied by a function, the derivative of the product is the constant multiplied by the derivative of the function itself.

To understand this better, consider the term \(2x\). Applying the constant rule, the derivative \(d/dx\) of \(2x\) is simply \(2\times d/dx(x)\). And we know the derivative of \(x\) is \(1\). Therefore:

• Calculate derivative of \(x\): \(d/dx(x) = 1\)
• Apply constant: \(2 \times 1 = 2\)

Similarly, for a constant term like \(1\), the derivative is \(0\) because constants do not change. Always remember that multiplying by a constant simplifies the differentiation process.

Trigonometric Derivatives

Trigonometric derivatives are crucial for understanding and working with functions involving trigonometric identities, such as sine, cosine, and tangent. They have specific established rules for differentiation.

In our equation, we encountered the term \(\cos x\). The derivative of \(\cos x\) is \(-\sin x\). Using this rule, we differentiate \(4\cos x\) by applying both the trigonometric derivative and the constant rule, resulting in:

• Differentiate \(\cos x\): \(d/dx(\cos x) = -\sin x\)
• Multiply by the constant: \(4(-\sin x) = -4\sin x\)

Understanding these trigonometric derivatives allows you to solve a wide range of calculations involving periodic and oscillating functions. Each trigonometric function has its own specific derivative, which helps simplify various mathematical problems.