Problem 18 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\exp [\cos (4 x)]\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The given function is \( f(x) = \exp[\cos(4x)] \). This can be rewritten as \( f(x) = e^{\cos(4x)} \). We need to find its derivative with respect to \( x \).

Step 2: Apply the Chain Rule

To differentiate \( e^{\cos(4x)} \), use the chain rule. The derivative of \( e^u \) with respect to \( u \) is \( e^u \), so for \( e^{\cos(4x)} \), apply the chain rule: \( \frac{d}{dx}[e^{u}] = e^{u} \cdot \frac{du}{dx} \) where \( u = \cos(4x) \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( \cos(4x) \) with respect to \( x \): \( \frac{d}{dx}[\cos(4x)] = -\sin(4x) \cdot \frac{d}{dx}[4x] \). Since \( \frac{d}{dx}[4x] = 4 \), it follows that \( \frac{d}{dx}[\cos(4x)] = -4\sin(4x) \).

Step 4: Combine Results from Chain Rule

Apply the chain rule using the results from Step 3: \[ \frac{d}{dx}[e^{\cos(4x)}] = e^{\cos(4x)} \times (-4\sin(4x)) \].

Step 5: Write the Final Derivative

The derivative of \( f(x) = e^{\cos(4x)} \) with respect to \( x \) is \( -4\sin(4x) e^{\cos(4x)} \).

Key Concepts

Chain Rule

The chain rule is a fundamental differentiation technique used to find the derivative of composite functions. When you have a function within a function, like in the expression \( f(x) = e^{\cos(4x)} \), the chain rule makes it possible to differentiate by focusing on each part separately. To apply the chain rule, we break down the differentiation process into two main components:

• Differentiate the outer function while keeping the inner function intact.
• Multiply by the derivative of the inner function.

In our function \( f(x) = e^{\cos(4x)} \), the outer function is \( e^u \) (where \( u = \cos(4x) \)) and the inner function is \( \cos(4x) \). The differentiation steps are:1. Differentiate \( e^u \) with respect to \( u \), resulting in \( e^u \).2. Then, multiply by the derivative of \( u \) (which is \( \cos(4x) \)) with respect to \( x \).This method allows us to handle more complex derivatives efficiently—making it a powerful tool in calculus.

Exponential Functions

Exponential functions are pervasive in mathematics because of their unique properties, especially when it comes to growth and decay processes. The general form of an exponential function is \( e^x \), where \( e \) is Euler's number, approximately \( 2.718 \).For functions like \( f(x) = e^{\cos(4x)} \), we are dealing with an exponential function that has another function plugged into the exponent. This gives rise to interesting behaviors and aligns perfectly with the chain rule to find its derivative.A key property of exponential functions is that the derivative of \( e^x \) is itself \( e^x \). This makes differentiating exponential functions straightforward when using the chain rule—a practice seen in solving \( \frac{d}{dx}[e^{\cos(4x)}] \).In essence, the exponential function's simplicity in differentiation, along with its ability to model dynamic processes, makes it an essential topic for students to understand.

Trigonometric Functions

Trigonometric functions, like sine and cosine, describe the relationships of angles in geometry and are especially vital in calculus for oscillatory behavior. Specifically, the cosine function \( \cos(x) \) is pivotal in our original exercise, as it creates a nested function situation in \( f(x) = e^{\cos(4x)} \).Understanding the derivatives of trigonometric functions is crucial:- The derivative of \( \cos(x) \) is \( -\sin(x) \).- When differentiating \( \cos(4x) \), we apply the chain rule again: first differentiate \( \cos(u) \) with respect to \( u \) to get \( -\sin(u) \), then multiply by the derivative of the inner function \( u = 4x \), which results in \( 4 \).These concepts come together in the differentiation process when calculating \( \frac{d}{dx}[\cos(4x)] \) as seen in the original exercise. By grasping these trigonometric properties, students can handle complex differentiation involving trigonometric and exponential composites effectively.