Problem 18 - Calculus for Biology and Medicine

Question

Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ f(x)=e^{\cos (4 x)} $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = -4e^{\cos(4x)} \sin(4x) \).

Step 1: Identify the Components

In the function \( f(x) = e^{\cos(4x)} \), we recognize that this is a composite function where the outer function is \( e^u \) and the inner function is \( u = \cos(4x) \). We need to apply the chain rule for differentiation.

Step 2: Differentiate the Outer Function

First, differentiate the outer function \( e^u \) with respect to \( u \). The derivative of \( e^u \) is \( e^u \), so the derivative of the outer part with respect to \( u \) is \( e^{\cos(4x)} \).

Step 3: Differentiate the Inner Function

Now, differentiate the inner function \( \cos(4x) \) with respect to \( x \). The derivative of \( \cos(x) \) is \( -\sin(x) \), and by the chain rule, the derivative of \( \cos(4x) \) is \( -\sin(4x) \cdot 4 = -4\sin(4x) \).

Step 4: Apply the Chain Rule

Using the chain rule, the derivative of \( f(x) = e^{\cos(4x)} \) is obtained by multiplying the derivative of the outer function \( e^{\cos(4x)} \) by the derivative of the inner function \( -4\sin(4x) \). This gives us: \( f'(x) = e^{\cos(4x)} \cdot (-4\sin(4x)) \).

Step 5: Simplify the Expression

Simplify the expression for the derivative: \( f'(x) = -4e^{\cos(4x)} \sin(4x) \). This is the final differentiated form of the function.

Key Concepts

Chain Rule

The chain rule is a fundamental tool in calculus for taking the derivative of composite functions. It allows us to differentiate complex expressions step-by-step. When you have a function within another function, you need to apply the chain rule to find the derivative.
Here's how it works:

• Identify the outer function and the inner function.
• Take the derivative of the outer function with respect to its inner function.
• Multiply by the derivative of the inner function with respect to the variable.

In the exercise, we used the chain rule on the function \( f(x) = e^{\cos(4x)} \). The outer function is \( e^u \) and the inner function is \( u = \cos(4x) \). By applying the chain rule, we successfully found the derivative.

Composite Functions

A composite function is formed when one function is applied to the results of another function. In simpler terms, it's a function within a function. For example, in the function \( e^{\cos(4x)} \),\( \cos(4x) \) is inside the exponential function \( e^u \).
Understanding composite functions is essential because it allows us to see where and how to apply the chain rule. The notation usually appears as \( f(g(x)) \), where \( f \) is the outer function and \( g \) is the inner function.
When tackling problems involving composites:

• First, identify the layers of functions involved.
• Recognize dependencies – which function is derived from inside another.
• Use techniques like the chain rule to differentiate effectively.

Derivative Calculation

Calculating derivatives involves applying rules and methods to find the rate of change of a function. This helps us understand how a function behaves as its input changes. When dealing with functions like \( f(x) = e^{\cos(4x)} \), derivative calculations become crucial for exploring complex interactions.
Here’s a simple guide:

• Identify the function type – whether it's simple or a composite one.
• Use basic rules: the power rule, product rule, and especially, the chain rule for composites.
• Simplify your result for clarity and easier interpretation.

In our solution, the differentiation involved simplifying \( f'(x) = -4e^{\cos(4x)} \sin(4x) \), which gives insights into the behavior of \( f(x) \). It tells us how fast \( f(x) \) changes and in which direction as \( x \) varies.

Trigonometric Functions

Trigonometric functions such as \( \cos(x) \) and \( \sin(x) \) are fundamental in calculus, especially when dealing with oscillatory or wave-like behaviors. Understanding how to differentiate these functions is crucial for solving trigonometric derivative problems.
For instance, the derivative of the cosine function \( \cos(x) \) is \( -\sin(x) \). When combined with linear transformations, like \( \cos(4x) \), we use the chain rule: multiply the derivative \( -\sin(4x) \) by the derivative of \( 4x \), which is 4, resulting in \( -4\sin(4x) \).
Key points when dealing with trigonometric derivatives:

• Remember basic trigonometric derivatives: \( \frac{d}{dx} \cos(x) = -\sin(x) \) and \( \frac{d}{dx} \sin(x) = \cos(x) \).
• Apply the chain rule when trigonometric functions have composite arguments.
• Understand how these derivatives impact the overall function and its graph.

In the given exercise, effectively differentiating \( \cos(4x) \) was vital to finding \( f'(x) \). This understanding also helps interpret real-world phenomena modeled by such functions.