Problem 90 - Calculus for Biology and Medicine

Question

Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y\) $$ y=\frac{x^{2}}{f(x)-g(x)} $$

Step-by-Step Solution

Answer

The derivative is \( y' = \frac{2x(f(x) - g(x)) - x^2(f'(x) - g'(x))}{(f(x) - g(x))^2} \).

Step 1: Identify the Function

We are given a function where \[ y = \frac{x^2}{f(x) - g(x)} \]We need to find the derivative of this function with respect to \(x\). This function is a quotient. We will apply the quotient rule to find the derivative.

Step 2: Apply the Quotient Rule

The quotient rule states that if \( y = \frac{u(x)}{v(x)} \), then the derivative is given by:\[ y' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \]For our function, let:- \( u(x) = x^2 \) so \( u'(x) = 2x \)- \( v(x) = f(x) - g(x) \) so \( v'(x) = f'(x) - g'(x) \)

Step 3: Compute the Derivative

Using the quotient rule, substitute the derivatives and functions into the formula:\[ y' = \frac{(2x)(f(x) - g(x)) - (x^2)(f'(x) - g'(x))}{(f(x) - g(x))^2} \]Simplify the expression as needed.

Step 4: Simplify the Expression

Distribute the terms in the numerator:\[ y' = \frac{2x(f(x) - g(x)) - x^2(f'(x) - g'(x))}{(f(x) - g(x))^2} \]There's no further simplification needed unless specific forms of \(f(x)\) or \(g(x)\) are known. This is the expression for the derivative of \(y\) with respect to \(x\).

Key Concepts

Quotient Rule

The quotient rule is an essential derivative rule in calculus. It allows us to differentiate functions that are fractions, where one function is divided by another. When you have a function in the form \( \frac{u(x)}{v(x)} \), the quotient rule helps us find the derivative efficiently.
The formal statement of the quotient rule is:

• If \( y = \frac{u(x)}{v(x)} \), then \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2} \).

This rule requires you to calculate the derivatives of both the numerator \( u(x) \) and the denominator \( v(x) \). Once you have \( u'(x) \) and \( v'(x) \), you substitute these into the formula. An example from our exercise shows the use of the quotient rule:

• Here, \( u(x) = x^2 \) and \( v(x) = f(x) - g(x) \).
• The derivatives are calculated as \( u'(x) = 2x \) and \( v'(x) = f'(x) - g'(x) \).

Substituting these into the formula provides the derivative of the function. This process makes it easier to handle complex fractions.

Differentiable Functions

In calculus, differentiable functions are those for which derivatives exist. A function \( f(x) \) is differentiable at a point \( x \) if its derivative \( f'(x) \) is defined there. Differentiability is a critical concept because it confirms the smoothness and continuity of functions at a given point.
For a function to be differentiable:

• The function must be continuous at that point.
• There must not be any sharp corners at the point.
• No discontinuities should exist in the neighborhood of that point.

In our exercise, both \( f(x) \) and \( g(x) \) are differentiable at \( x \). This means they have continuous derivatives, and the quotient rule can be safely applied to \( \frac{x^2}{f(x) - g(x)} \). Having differentiable functions ensures the derivatives can be calculated correctly, allowing for accurate calculus problem solving.

Calculus Problem Solving

Solving calculus problems, especially those concerning derivatives, is about understanding the rules and applying them step by step. The problem we considered involved a complex fraction, so the main idea was to use the quotient rule to solve it efficiently.
The steps in calculus problem solving generally include:

• Identifying the type of function and the correct rule to apply – whether it's the power rule, product rule, quotient rule, etc.
• Calculating the necessary derivatives accurately.
• Substituting these derivatives back into the relevant formula for derivation.
• Simplifying the result to get the derivative of the function.

In our specific problem, identifying the need to use the quotient rule was the key. Once that was clear, the steps flowed logically to produce the derivative of the function \( y = \frac{x^2}{f(x) - g(x)} \). When you practice problem solving in calculus, consistent application of these structured steps can simplify even the most complex problems.