Problem 66 - Calculus for Biology and Medicine

Question

Find the derivatives of the following functions: $$ f(x)=\cos \sqrt{x^{2}+1} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \cos \sqrt{x^2+1} \) is \( f'(x) = -\frac{x \sin(\sqrt{x^2+1})}{\sqrt{x^2+1}} \).

Step 1: Identify the Outer and Inner Functions

In the function \( f(x) = \cos \sqrt{x^2+1} \), the outer function is \( g(u) = \cos u \) and the inner function is \( h(x) = \sqrt{x^2+1} \). Identify these so that the chain rule can be applied to find the derivative.

Step 2: Differentiate the Outer Function

The derivative of the outer function \( g(u) = \cos u \) with respect to \( u \) is \( g'(u) = -\sin u \). This derivative will be used in the application of the chain rule.

Step 3: Differentiate the Inner Function

The inner function \( h(x) = \sqrt{x^2+1} \) can be rewritten as \( (x^2+1)^{1/2} \). The derivative of \( h(x) \) with respect to \( x \) is found using the chain rule and is \( h'(x) = \frac{x}{\sqrt{x^2+1}} \).

Step 4: Apply the Chain Rule

According to the chain rule, \( f'(x) = g'(h(x)) \cdot h'(x) \). Substitute the derivatives from Steps 2 and 3: \[ f'(x) = -\sin(\sqrt{x^2+1}) \cdot \frac{x}{\sqrt{x^2+1}} \]

Step 5: Simplify the Expression

Simplify the expression for the derivative by combining terms:\[ f'(x) = -\frac{x \sin(\sqrt{x^2+1})}{\sqrt{x^2+1}} \]

Key Concepts

Understanding Derivatives

The concept of derivatives is a fundamental aspect of calculus. In simple terms, a derivative represents the rate at which a function changes at any given point. Think of it as measuring the slope of a curve at a specific point. In our exercise, we are asked to find the derivative of the function \( f(x) = \cos \sqrt{x^2+1} \). To compute this derivative, we must examine how the function changes as \( x \) varies. This involves determining the slopes of two interrelated functions, known as the outer and inner functions. Derivatives allow us to understand the intricate relationships and dynamics within these layered functions. It provides insight into their behaviors and can highlight trends and patterns.
Key elements to remember about derivatives:

• They tell us how functions are changing.
• Provide insights into function behavior and direction.
• Essential for applying calculus to real-world problems.

Exploring Outer and Inner Functions

In the chain rule process, identifying the outer and inner functions is crucial. The chain rule is a method for differentiating composite functions. A composite function is one where two functions are nested inside each other.
- **Outer Function:** For our function \( f(x) = \cos \sqrt{x^2+1} \), the outer function is \( g(u) = \cos u \). This function takes the inner function as its input.
- **Inner Function:** The inner function in this case is \( h(x) = \sqrt{x^2+1} \), which we can rewrite as \( (x^2 + 1)^{1/2} \). This function feeds its output into the outer function.
Recognizing which is the inner and outer function allows us to apply the chain rule effectively. The chain rule states that to differentiate a composite function, we first differentiate the outer function and then multiply it by the derivative of the inner function. This step-by-step approach is central to solving calculus problems involving layers of functions.

Simplifying the Function

Function simplification is an integral step after applying the chain rule in derivative calculations. Once we've applied the chain rule to our function, simplifying the resultant expression makes it cleaner and easier to interpret. In the case we're dealing with, after applying the chain rule, we arrive at:
\[ f'(x) = -\sin(\sqrt{x^2+1}) \cdot \frac{x}{\sqrt{x^2+1}} \]
By simplifying, we combine like terms to reach:
\[ f'(x) = -\frac{x \sin(\sqrt{x^2+1})}{\sqrt{x^2+1}} \]
Simplification involves:

• Combining fractions and like terms.
• Reducing complex expressions to their simplest form.
• Providing a more digestible result for deeper analysis.

Through simplification, the derivative becomes more manageable and ready for direct application or further mathematical exploration.