Problem 4 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=-\sin x+\cos x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Components

In the function given, \( f(x) = -\sin x + \cos x \), we have two trigonometric components: \(-\sin x\) and \(\cos x\). We'll address each separately in the differentiation process.

Step 2: Differentiate \(-\sin x\)

The derivative of \(-\sin x\) with respect to \(x\) is \( -\cos x \), since the derivative of \( \sin x \) is \( \cos x \) and the negative sign is preserved.

Step 3: Differentiate \(\cos x\)

The derivative of \(\cos x\) with respect to \(x\) is \(-\sin x\), as the derivative of \(\cos x\) is \(-\sin x\).

Step 4: Combine the Derivatives

Combine the derivatives of each component: \(-\cos x + (-\sin x)\), which simplifies to \(-\cos x - \sin x\). This gives us the derivative of the original function.

Key Concepts

Trigonometric Differentiation

Trigonometric differentiation is the process of finding the derivative of functions that involve trigonometric terms such as \(\sin x\), \(\cos x\), \(\tan x\), and their inverses. It's a vital part of calculus, especially when dealing with periodic or oscillating functions.

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

When differentiating a function like \(-\sin x + \cos x\), trigonometric derivatives are applied to each component. Each term is handled separately followed by combining their respective derivatives. This method helps in dissecting complex trigonometric functions into manageable parts, making the differentiation process straightforward.

Chain Rule

Even though the exercise here doesn’t directly use the chain rule, understanding this principle is essential in calculus. The chain rule is used to differentiate compositions of functions, i.e., functions inside of other functions. It is expressed in a straightforward manner: if you have a function \(y = f(g(x))\), the derivative \(dy/dx\) is found by multiplying the derivative of the outer function by the derivative of the inner function. Mathematically, it's represented as: \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\). Suppose you have a trigonometric function that is composed, for example, \(\sin(2x)\). The chain rule tells us that we first differentiate \(\sin\) with respect to its argument \(2x\), giving \(\cos(2x)\), and then multiply by the derivative of \(2x\), which is \(2\). Thus, the derivative of \(\sin(2x)\) is \(2\cos(2x)\).The chain rule is a powerful tool, allowing us to break down complex derivatives step by step.

Calculus Process

The calculus process for finding derivatives involves several important steps, which can be broken down to simplify complex problems. These steps include identifying the components, applying differentiation rules, and combining results effectively. Each derivative problem has its unique aspects, but the process remains largely consistent.1. **Identify the terms** - First, break down the function into its basic components. In our example, the function \(f(x) = -\sin x + \cos x\) is divided into two trigonometric parts. 2. **Apply derivatives** - Use known differentiation rules, like those for trigonometric functions, to find the derivative of each term separately. 3. **Combine results** - Once each part is differentiated, combine them to get the complete derivative of the function. This process is linear and helps ensure accuracy by working through each element individually.The calculus process is systematic, making tasks like trigonometric differentiation more approachable, and it serves as a foundation for solving a wide variety of problems across calculus.