Problem 75 - Calculus for Biology and Medicine

Question

Differentiate $$ f(x)=\frac{a x}{3+x} $$ with respect to \(x\). Assume that \(a\) is a positive constant.

Step-by-Step Solution

Answer

The derivative is \( \frac{3a}{(3+x)^2} \).

Step 1: Apply the Quotient Rule

The function you are differentiating is a quotient \( \frac{u}{v} \) where \( u = a x \) and \( v = 3 + x \). The quotient rule states \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \).

Step 2: Differentiate the Numerator

Differentiate \( u = a x \) with respect to \( x \). The derivative of \( a x \) is \( u' = a \) since \( a \) is a constant multiplier of \( x \).

Step 3: Differentiate the Denominator

Differentiate \( v = 3 + x \) with respect to \( x \). The derivative of \( 3 + x \) is \( v' = 1 \) because the derivative of a constant is zero and the derivative of \( x \) with respect to \( x \) is one.

Step 4: Plug into the Quotient Rule Formula

Substitute \( u' = a \), \( v = 3 + x \), \( u = a x \), and \( v' = 1 \) into the quotient rule: \[ \left( \frac{a x}{3 + x} \right)' = \frac{a(3 + x) - a x (1)}{(3 + x)^2} \].

Step 5: Simplify the Expression

Simplify the expression obtained from applying the quotient rule: \[ \frac{a(3 + x) - a x}{(3 + x)^2} = \frac{3a}{(3 + x)^2} \]. Cancel out \( a x \) from the terms in the numerator.

Key Concepts

Quotient Rule

The Quotient Rule is a helpful technique in calculus, used to differentiate functions in the form of a fraction. This rule is necessary when a function is expressed as the quotient of two differentiable functions, say \( \frac{u}{v} \), where \( u \) and \( v \) are functions of \( x \).

To differentiate such functions, the quotient rule formula is stated as:

• \( \left( \frac{u}{v} \right)' = \frac{u'v - uv'}{v^2} \)

Here, \( u' \) and \( v' \) represent the derivatives of \( u \) and \( v \) with respect to \( x \).

This rule helps to systematically apply differentiation to both the numerator and the denominator, simplifying more complex expressions that arise when rates of change are not uniform across a ratio.

Understanding the steps involved in applying the quotient rule makes it easier for learners to break down complex expressions into more manageable parts.

Derivative

The concept of a derivative is fundamental in calculus. It is used to determine the rate of change of a function with respect to a variable. For instance, when we consider a function \( f(x) \), its derivative, denoted as \( f'(x) \), shows how \( f \) changes as \( x \) changes.

In simpler terms, the derivative gives us a snapshot of the function's slope at any given point. A great way to visualize this is to think of the function as a curve and the derivative as the slope of the tangent line at any point on this curve.

Basic power rule, chain rule, and our main focus here, the quotient rule, are techniques used to find the derivatives in different scenarios:

• Power Rule: For a function \( f(x) = x^n \), the derivative is \( f'(x) = n \cdot x^{n-1} \)
• Quotient Rule: Used for functions that are ratios of two functions.

The derivative not only tells us the slope of the tangent line but also can indicate where the function is increasing or decreasing and where local maximum and minimum points occur.

Calculus

Calculus is the branch of mathematics that deals with the study of rates of change and accumulation. It is foundational for various fields of study including physics, engineering, economics, and beyond.

The subject divides into two main branches: Differential Calculus and Integral Calculus.

• Differential Calculus focuses on the concept of the derivative, which, as mentioned, is the rate of change of quantities.
• Integral Calculus, on the other hand, deals with accumulation and area under curves, primarily using integration techniques.

Together, these subjects allow mathematicians and scientists to model real-world scenarios where dynamic changes are present, such as calculating trajectories, predicting economic trends, or assessing biological growth.

Each technique in calculus, including the quotient rule discussed here, supports the broader goal of understanding and describing change in various contexts. It provides insight into the behavior of functions and is indispensable in not only theoretical mathematics but practical problem-solving across countless industries.