Problem 95 - Calculus for Biology and Medicine

Question

In this problem we will prove the quotient rule using an argument similar to the one used to prove the product rule in Section 4.4.1. Let \(u(x)\) and \(v(x)\) be differentiable functions, and define a quotient function \(f(x)=\frac{u(x)}{v(x)} .\) The derivative \(f^{\prime}(x)\), if it exists, is equal to: $$ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} $$ (a) Assuming \(v(x) \neq 0\), show that the quotient on the right-hand side can be written as: \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(u(x+h) v(x)-u(x) v(x))-(u(x) v(x+h)-u(x) v(x))}{h v(x) v(x+h)}\) and then rearranged into: $$ f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\left(\frac{u(x+h)-u(x)}{h}\right) v(x)-u(x)\left(\frac{u(x+h)-u(x)}{h}\right)}{v(x) v(x+h)} $$ (b) Using the limit laws, show that the equation from (a) can be rewritten as $$ f^{\prime}(x)=\frac{\left(\lim _{h \rightarrow 0} \frac{u(x+h)-u(x)}{h}\right) v(x)-u(x)\left(\lim _{h \rightarrow 0} \frac{v(x+h)-v(x)}{h}\right)}{v(x) \lim _{h \rightarrow 0}(v(x+h))} $$ provided all of the limits exist, and provided \(\lim _{h \rightarrow 0} v(x+h) \neq 0\). (c) Using the formal definition of a derivative write \(f^{\prime}(x)\) in terms of \(u(x), v(x), u^{\prime}(x)\), and \(v^{\prime}(x) .\) Hence prove the quotient law.

Step-by-Step Solution

Answer

The derivative of \(f(x) = \frac{u(x)}{v(x)}\) is \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}\). This is proven using the difference quotient and limit laws.

Step 1: Simplify the Difference Quotient for the Function

The problem gives us the quotient function as \(f(x) = \frac{u(x)}{v(x)}\), and we need to find \(f'(x)\) using the definition of the derivative:\[f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\]Substitute \(f(x)\) and \(f(x+h)\) into the quotient:\[f(x+h) = \frac{u(x+h)}{v(x+h)},\quad f(x) = \frac{u(x)}{v(x)}\]Thus,\[f'(x) = \lim_{h \to 0} \frac{\frac{u(x+h)}{v(x+h)} - \frac{u(x)}{v(x)}}{h}\]Next, use the common denominator to simplify this:\[= \lim_{h \to 0} \frac{u(x+h) v(x) - u(x) v(x+h)}{h v(x) v(x+h)}\]

Step 2: Rearrange the Expression Using the Difference Quotient for u(x) and v(x)

In this step, we rearrange the expression by separating terms associated with \(u(x)\) and \(v(x)\):\[f'(x) = \lim_{h \to 0} \frac{(u(x+h) - u(x))v(x) - u(x)(v(x+h) - v(x))}{hv(x)v(x+h)}\]Recognize each part as a difference quotient. The numerator can be expanded as:\[\left(\frac{u(x+h) - u(x)}{h}\right)v(x) - u(x)\left(\frac{v(x+h) - v(x)}{h}\right)\]

Step 3: Apply Limit Laws to Factor Out Derivatives

Using the definition of derivatives, apply limit laws to factor derivatives out of the difference quotient:\[f'(x) = \frac{\left(\lim_{h \to 0} \frac{u(x+h) - u(x)}{h}\right) v(x) - u(x) \left(\lim_{h \to 0} \frac{v(x+h) - v(x)}{h}\right)}{v(x) \lim_{h \to 0} v(x+h)}\]Here:- \(\lim_{h \to 0} \frac{u(x+h) - u(x)}{h} = u'(x)\)- \(\lim_{h \to 0} \frac{v(x+h) - v(x)}{h} = v'(x)\)- Ensure \(\lim_{h \to 0} v(x+h) = v(x)\) and \(v(x) eq 0\) since \(v(x+h)\) approaches \(v(x)\).

Step 4: Substitute Derivatives into the Expression

Substitute the derivatives into the expression derived:\[f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}\]This equation is the quotient rule, where the derivation results in the numerator representing the difference in changes weighted by the other function, and the denominator being the square of the denominator function \(v(x)\).

Key Concepts

Derivative

A derivative represents how a function changes as its input changes. It is a fundamental concept in calculus that helps determine the rate of change or the slope of the function at any given point. For a function \( f(x) \), the derivative, denoted as \( f'(x) \), is mathematically defined as: \[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]This formula is called the difference quotient. It calculates the instantaneous rate of change by taking the limit of the average rate of change as the interval \( h \) approaches zero.

In the context of the quotient rule, we use this definition to find the derivative of a quotient of two functions \( \frac{u(x)}{v(x)} \). It's a tool for understanding how combined functions behave in their differentiation.

Limit Laws

Limit laws are essential mathematical rules that enable us to evaluate limits with greater ease. They're particularly useful when dealing with complex expressions. The fundamental limit laws used in calculus include:

• Sum Law: \( \lim_{x \to a} (f(x) + g(x)) = \lim_{x \to a} f(x) + \lim_{x \to a} g(x) \)
• Product Law: \( \lim_{x \to a} (f(x) \cdot g(x)) = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \)
• Quotient Law: \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} \), provided \( \lim_{x \to a} g(x) eq 0 \)

In proving the quotient rule, we heavily rely on these laws. We rearrange and manipulate expressions, then apply the limit laws to factor out derivatives. This approach simplifies complex fractions and confirms the existence of limits, ensuring our derivations are mathematically sound.

Differentiable Functions

A function is said to be differentiable at a point if it has a derivative at that point. This implies not just that the function changes, but that this change is smooth and continuous. In mathematical terms, a function \( f(x) \) is differentiable at \( x = a \) if:\[ \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \]exists.

When functions \( u(x) \) and \( v(x) \) are differentiable, it means we can apply the quotient rule to \( \frac{u(x)}{v(x)} \) as long as \( v(x) eq 0 \). This rule, which uses the property of differentiability, highlights the smooth behavior and calculability of complex expressions. Differentiability is a key requirement in calculus, enabling efficient computation of derivatives for functions combined via arithmetic operations.

Formal Definition of Derivative

The formal definition of a derivative lays the groundwork for understanding how derivatives operate. It can be thought of as a standard formula that explains how changes in output arise from changes in input for a function. For a function \( f(x) \), the derivative is formally defined as:\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]This foundational concept plays a crucial role in proving the quotient rule.

In the quotient rule scenario, for a function defined as the ratio of \( u(x) \) over \( v(x) \), each part of our derivation utilizes this definition. The systematic rearrangement of the expression reflects the formal definition and the properties of limits to arrive at:\[ f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2} \]Ultimately, the formal definition encapsulates the underpinning logic we'll employ and apply to articulate and substantiate the rule for differentiating quotients.