Problem 55 - Calculus for Biology and Medicine

Question

Find the derivative with respect to the independent variable. $$ f(x)=\tan \frac{1}{x} $$

Step-by-Step Solution

Answer

The derivative is \(-\frac{\sec^2\left(\frac{1}{x}\right)}{x^2}\)."

Step 1: Identify the Composition

The function given is a composition of two functions: the tangent function and the reciprocal function. We express it as: \[ f(x) = \tan(g(x)) \] where \[ g(x) = \frac{1}{x} \].

Step 2: Use the Chain Rule

The derivative of a composite function \( f(g(x)) \) is given by the chain rule:\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \].For our function, \( f(x) = \tan(g(x)) \), this becomes:\[ \frac{d}{dx}[\tan\left(\frac{1}{x}\right)] = \sec^2\left(\frac{1}{x}\right) \cdot g'(x) \].

Step 3: Differentiate the Inner Function

The inner function \( g(x) = \frac{1}{x} \) can be rewritten as \( g(x) = x^{-1} \). Its derivative is:\[ g'(x) = \frac{d}{dx}(x^{-1}) = -x^{-2} = -\frac{1}{x^2} \].

Step 4: Combine the Derivatives

Plug the derivative of the inner function back into the chain rule expression:\[ \frac{d}{dx}[\tan\left(\frac{1}{x}\right)] = \sec^2\left(\frac{1}{x}\right) \cdot (-\frac{1}{x^2}) \].Therefore, the derivative is:\[ -\frac{\sec^2\left(\frac{1}{x}\right)}{x^2} \].

Key Concepts

Chain Rule in Calculus

The chain rule is a fundamental principle in calculus, especially when dealing with composite functions. A composite function is one that is formed by combining two or more functions. For instance, in the original exercise, there was a combination of the trigonometric function tangent and the reciprocal function. To find the derivative of such a function, we apply the chain rule.

The chain rule states that if you have a function, say \( f(g(x)) \), where \( f \) and \( g \) are both functions of \( x \), the derivative is:

• \( \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \)

In simpler terms, you differentiate the outer function \( f \) with respect to \( g(x) \), multiply by the derivative of the inner function \( g(x) \).

For example, in our exercise, the inner function was \( g(x) = \frac{1}{x} \), and it was necessary to first find this derivative before applying it to the entire expression. This powerful tool allows us to tackle complex problems in calculus efficiently, which is vital for students studying calculus for various fields, such as biology or engineering.

Understanding Trigonometric Functions

Trigonometric functions are essential in mathematics. They describe the relationships between the angles and sides of triangles and are widely used in various scientific areas. These functions include sine, cosine, and tangent, among others.

In the given exercise, we focus on the tangent function. The basic property of the tangent function is that it is periodic and relates to the sine and cosine functions through the expression \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).

• The derivative of \( \tan(x) \) is \( \sec^2(x) \), an important derivative that pops up frequently.
• \( \sec(x) \) is another trigonometric function, defined as \( \sec(x) = \frac{1}{\cos(x)} \).

When dealing with derivatives, these relationships allow us to transform and simplify our expressions, as shown in the solution.

In calculus for biology, trigonometric functions can be instrumental in modeling periodic physiological processes, like heartbeats or circadian rhythms, making understanding of their calculus transformations crucial.

Calculus for Biology Applications

Calculus is not only a crucial mathematical tool but also plays a vital role in biological studies. In biology, we often use calculus to model dynamic processes and understand natural phenomena. Understanding derivatives is particularly important, as they describe rates of change, a common occurrence in biology.

For example, in the context of population dynamics, derivatives can be used to model how populations grow or shrink over time. Similarly, they can describe how a drug concentration changes within a human body.

The derivative in our original exercise demonstrates how changes in one variable affect another, a principle that is directly applicable to analyzing biological data. By comprehending how to calculate derivatives using methods like the chain rule, students can apply these skills to interpret various biological systems. This fusion of calculus and biology empowers students to make predictions, solve real-world problems, and develop a deeper understanding of life processes.