Problem 76 - Calculus for Biology and Medicine

Question

Differentiate $$ f(x)=\frac{a x}{k+x} $$ with respect to \(x\). Assume that \(a\) and \(k\) are positive constants.

Step-by-Step Solution

Answer

The derivative is \(f'(x) = \frac{a k}{(k + x)^2}\).

Step 1: Identify the Rule

The given function is a quotient of two functions: the numerator is \(a x\) and the denominator is \(k + x\). To differentiate this, we'll use the Quotient Rule, which states: if \(u(x)\) and \(v(x)\) are functions of \(x\), then the derivative of \(\frac{u(x)}{v(x)}\) is \(\frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).

Step 2: Define the Functions

Let \(u(x) = a x\) and \(v(x) = k + x\). We need to differentiate both \(u(x)\) and \(v(x)\). We will also square the denominator \(v(x)\).

Step 3: Differentiate \(u(x)=a x\)

The derivative \(u'(x)\) is the derivative of \(a x\), which is simply \(a\), because the derivative of \(x\) with respect to \(x\) is 1, and \(a\) is a constant.

Step 4: Differentiate \(v(x)=k+x\)

The derivative \(v'(x)\) is the derivative of \(k + x\). Since \(k\) is a constant, its derivative is 0, and the derivative of \(x\) is 1. Hence, \(v'(x) = 1\).

Step 5: Apply the Quotient Rule

Substitute the derivatives into the quotient rule formula: \[ f'(x) = \frac{a(k + x) - a x\cdot 1}{(k + x)^2} \].

Step 6: Simplify the Expression

Simplify the numerator: \(a(k + x) - a x = a k\). Thus, the derivative simplifies to \[ f'(x) = \frac{a k}{(k + x)^2} \].

Key Concepts

Quotient Rule

Differentiation is a way to measure how a function changes as its input changes, and when we deal with functions that are ratios of other functions, we use the Quotient Rule. This rule simplifies the process of finding derivatives of such function fractions, where one function divides another. Let's break down how this rule is applied.

• You first need two functions: a numerator function, which we'll call \( u(x) \), and a denominator function, which we'll call \( v(x) \).
• The formula to find the derivative of \( \frac{u(x)}{v(x)} \) is:
  \[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \]
• This formula tells us to first differentiate \( u(x) \) to get \( u'(x) \), and \( v(x) \) to get \( v'(x) \), and then plug these derivatives into the formula.

This process allows you to compute the derivative of a quotient function accurately, which is particularly useful in cases like the exercise where the function is expressed as a fraction.

Derivative

A derivative represents the rate of change of a function with respect to its variable. In simpler terms, it measures how much the output of a function changes as the input changes slightly. This concept is crucial in understanding how functions behave dynamically.

• The derivative of a function \( f(x) \) at a point \( x \) is denoted by \( f'(x) \).
• For basic functions, finding the derivative often involves simple rules like the power rule (for power functions) or constant factor rule (when multiplying by constants).

In the context of our exercise, we found the derivative of the function \( a x \), which is a linear function. Linear functions, like \( a x \), have a constant derivative, which is just \( a \) here because the rate of change of \( x \) with respect to itself is 1.

Functions

Understanding functions is key to grasping differentiation and applying rules like the Quotient Rule. A function is a relationship between a set of inputs and outputs, where each input is related to exactly one output.

• The input of a function is typically represented by \( x \), and the output is \( f(x) \).
• In our problem, we had the expression \( f(x) = \frac{a x}{k + x} \), which is a fraction made by two component functions: \( ax \) (the numerator) and \( k + x \) (the denominator).

When dealing with such functions, particularly those expressed as fractions, it's important to remember that each part can be treated as a separate smaller function. Approaching it this way makes differentiation more straightforward, as you can independently find each derivative and utilize them in conjunction with rules like the Quotient Rule.