Problem 13 - Calculus for Biology and Medicine

Question

In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=2 \sec (1+2 x) $$

Step-by-Step Solution

Answer

\( 4 \sec(1+2x)\tan(1+2x) \)

Step 1: Identify the outer and inner functions

In the function \( f(x) = 2 \sec (1+2x) \), identify the outer function and the inner function. Here, the outer function is \( 2 \sec(u) \) where \( u = 1+2x \).

Step 2: Derive the outer function

The derivative of \( \sec(u) \) with respect to \( u \) is \( \sec(u)\tan(u) \). So, the derivative of the outer function \( 2\sec(u) \) is \( 2\sec(u)\tan(u) \).

Step 3: Derive the inner function

The inner function is \( u = 1 + 2x \). The derivative of \( 1 + 2x \) with respect to \( x \) is simply \( 2 \).

Step 4: Apply the Chain Rule

According to the Chain Rule, the derivative of a composite function \( f(g(x)) \) is \( f'(g(x)) \cdot g'(x) \). Substitute, and you get: \[ \frac{d}{dx}[2 \sec(1+2x)] = 2\sec(1+2x)\tan(1+2x) \cdot 2 \] This simplifies to:\[ 4 \sec(1+2x)\tan(1+2x) \].

Key Concepts

Understanding Derivatives

Derivatives in calculus are essentially used to comprehend how a function behaves at any given point. They tell us the rate at which a function is changing with respect to one of its variables. For instance, if you're driving a car, the speed of the car would be the derivative of the car's position with respect to time.

In mathematical terms, the derivative of a function \( f(x) \) is often expressed as \( f'(x) \) or \( \frac{df}{dx} \). It symbolizes the instantaneous rate of change. Computing derivatives is pivotal in analysis and understanding of functions in calculus.

When dealing with more complex functions, such as those involving trigonometric expressions or composite functions, it's crucial to identify how these parts connect to compute the derivative efficiently. This is where specific rules, like the chain rule, come into play to make the differentiation process manageable.

Decoding the Chain Rule

The Chain Rule is a powerful tool in calculus used to differentiate composite functions. A composite function is when one function is nested inside another, such as \( f(g(x)) \). The Chain Rule states that the derivative of this composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function itself.

Mathematically, this is represented as:

• \((f(g(x)))' = f'(g(x)) \cdot g'(x)\)

This rule is instrumental when dealing with complex functions because it simplifies finding their derivatives. When applied correctly, it breaks down the differentiation process into manageable parts, enabling you to focus on each function individually before combining results.

In our exercise with \( f(x) = 2 \sec(1+2x) \), the Chain Rule was used to find a final derivative of \( 4 \sec(1+2x)\tan(1+2x) \). This illustrates how multiple layers of a function can be differentiated effectively.

Trigonometric Functions and Their Derivatives

Trigonometric functions, such as sine, cosine, and secant, frequently appear in calculus problems, and understanding how to differentiate them is crucial. Each of these functions has a standard derivative, which is essential knowledge for anyone studying calculus.

For example, the derivative of \( \sec(x) \) is \( \sec(x)\tan(x) \). This derivative is derived from the relation between secant and cosine functions. These derivatives play a key role in problems involving trigonometric functions, especially when they are part of a composite function, as they determine how these functions' outputs change.

• \( \frac{d}{dx} [\sec(x)] = \sec(x) \tan(x) \)
• \( \frac{d}{dx} [\sin(x)] = \cos(x) \)
• \( \frac{d}{dx} [\cos(x)] = -\sin(x) \)

In the given exercise, recognizing that the derivative of \( \sec(u) \) is \( \sec(u)\tan(u) \) allowed us to apply the chain rule successfully when \( u = 1+2x \). This understanding ensures we effectively work through trigonometric-related calculus problems.

Exploring Composite Functions

Composite functions occur when one function is applied to the result of another function, essentially nesting them. Formally, if you have two functions, \( f \) and \( g \), the composite function is written as \( f(g(x)) \). In these scenarios, differentiation can seem daunting, but the chain rule simplifies it by treating the inner and outer functions individually.

To identify a composite function, look for functions within functions. For instance, in \( f(x) = 2 \sec(1+2x) \), the outer function is \( 2 \sec(u) \) and the inner function \( u \) is \( 1+2x \). The main approach is to:

• Differentiate the outer function.
• Differentiate the inner function.
• Apply the chain rule to combine these results.

By breaking down functions into simpler parts, composite functions become less intimidating, and this structured approach enables easier differentiation. Understanding the nuances of composite functions and utilizing rules like the chain rule is essential for tackling advanced calculus problems effectively.