Problem 11 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=\tan (4 x) $$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = 4 \sec^2(4x) \).

Step 1: Identify the Outer Function and Inner Function

In the function \( f(x) = \tan (4x) \), identify the outer function as \( \tan(u) \) and the inner function as \( u = 4x \). This helps in applying the chain rule, where the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Step 2: Differentiate the Outer Function

Differentiate the outer function \( \tan(u) \) with respect to \( u \). The derivative of \( \tan(u) \) is \( \sec^2(u) \). Substitute back the inner function: \( \sec^2(4x) \).

Step 3: Differentiate the Inner Function

Differentiate the inner function \( 4x \) with respect to \( x \). The derivative of \( 4x \) is \( 4 \).

Step 4: Apply the Chain Rule

Apply the chain rule to find the derivative of \( f(x) = \tan(4x) \). Multiply the derivative of the outer function \( \sec^2(4x) \) by the derivative of the inner function \( 4 \). Thus, the derivative is \( f'(x) = 4 \sec^2(4x) \).

Step 5: Write the Final Derivative

Combine the results from the previous steps to express the derivative of the function. The derivative of \( f(x) = \tan(4x) \) with respect to \( x \) is \( f'(x) = 4 \sec^2(4x) \).

Key Concepts

The Chain Rule

The chain rule is a fundamental technique in calculus for finding the derivative of composite functions. A composite function is when you have a function within another function, like wrapping one present inside another. The chain rule helps us "unwrap" these layers to find how changing one variable affects the overall function.

To use the chain rule, you follow these steps:

• Identify the outer function and the inner function in your composite function.
• Differentiate the outer function while keeping the inner function intact for now.
• Differentiate the inner function with respect to the independent variable.
• Multiply the derivative from the outer function by the derivative of the inner function.

In our example, with the function \( f(x) = \tan(4x) \):

• The outer function is \( \tan(u) \), and the inner function is \( u = 4x \).
• The derivative of \( \tan(u) \) is \( \sec^2(u) \).
• The derivative of \( 4x \) is \( 4 \).
• Thus, applying the chain rule gives us \( f'(x) = 4 \sec^2(4x) \).

This demonstrates how the chain rule effectively breaks down the differentiation process.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are essential in many areas of mathematics and science. They describe relationships in right-angled triangles and periodic phenomena.

• Tangent (\( \tan \)): The tangent function, used in our example, is the ratio of the opposite side to the adjacent side in a right-angled triangle. It describes the slope of angles in trigonometry.
• Secant (\( \sec \)): The secant function is the reciprocal of the cosine function. Mathematically, \( \sec(u) = \frac{1}{\cos(u)} \). In calculus, the function \( \sec^2(u) \) is important as it arises naturally when differentiating tangents.
• Trigonometric functions are periodic, meaning they repeat their values in regular intervals. This makes them particularly useful for modeling wave-like behaviors.

In differentiating \( \tan(4x) \), we utilize the derivative \( \sec^2(u) \) because of the unique behaviors of tangent and secant functions.

Understanding these trigonometric functions is crucial because they often appear in composite functions and help describe various physical phenomena like sound waves or circular motion.

Function Composition

Function composition occurs when you apply one function to the results of another, essentially "plugging" one function into another. This concept is important for applying the chain rule and commonly appears in calculus.

In the function \( f(x) = \tan(4x) \), the composition stems from having \( \tan \) as the outer function and \( 4x \) as the inner function. Function composition can be thought of as creating a "function sandwich," with one layer affecting the next.

• Recognize the outer function and apply its derivative while keeping the inner part intact initially. Here, the outer function is \( \tan(u) \).
• Identify the inner function, such as \( u = 4x \), affecting how the entire function behaves.

This layered approach simplifies the process of finding derivatives for more complex functions, revealing how each component influences the result.

Effective use of function composition allows for a systematic approach to handling complex calculus problems, illustrating the interconnectedness of different functions and their derivatives.