Problem 88 - 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 in terms of \(f(x), g(x), f^{\prime}(x)\), and \(g^{\prime}(x) .\) \(y=\frac{2 f(x)+x}{3 g(x)}\)

Step-by-Step Solution

Answer

The derivative is \( y' = \frac{6g(x)f'(x) + 3g(x) - 6g'(x)f(x) - 3xg'(x)}{9g(x)^2} \).

Step 1: Identify the Quotient Rule Formula

To differentiate a quotient, use the quotient rule: if you have a function \( y = \frac{u}{v} \), then \( y' = \frac{u'v - uv'}{v^2} \). In this problem, identify \( u = 2f(x) + x \) and \( v = 3g(x) \).

Step 2: Differentiate the Numerator

The numerator of the function is \( u = 2f(x) + x \). Differentiating with respect to \( x \) gives the derivative: \( u' = 2f'(x) + 1 \), since the derivative of \( 2f(x) \) is \( 2f'(x) \) and the derivative of \( x \) is \( 1 \).

Step 3: Differentiate the Denominator

The denominator of the function is \( v = 3g(x) \). Differentiating with respect to \( x \) gives the derivative: \( v' = 3g'(x) \), since the derivative of \( 3g(x) \) is \( 3g'(x) \).

Step 4: Apply the Quotient Rule

Plug \( u \, (2f(x) + x) \), \( u' \, (2f'(x) + 1) \), \( v \, (3g(x)) \), and \( v' \, (3g'(x)) \) into the quotient rule: \[ y' = \frac{(2f'(x) + 1)(3g(x)) - (2f(x) + x)(3g'(x))}{(3g(x))^2} \].

Step 5: Simplify to Get Final Expression

Expand and simplify the expression for the derivative: \[ y' = \frac{6g(x)f'(x) + 3g(x) - 6g'(x)f(x) - 3xg'(x)}{9g(x)^2} \]. Simplifying further if possible may help in specific contexts, but this form expresses the derivative in terms of \( f(x) \), \( g(x) \), \( f'(x) \), and \( g'(x) \).

Key Concepts

Differential Calculus

Differential calculus is all about understanding how things change. At its core, it's the study of derivatives — the rates at which things go up or down. Imagine you're climbing a hill. Differential calculus helps you decide if you're gaining height slowly or speedily at any given moment.
The secret weapon in differential calculus is the derivative. Think of it as a mathematical tool that tells you about change. When you differentiate a function, you're finding out how responsive it is to changes in its input.

• Change clue: Derivatives help reveal how sensitive a function is to changes.
• Real-world answers: It finds out things like how quickly cars speed up or the rate at which water flows into a tank.

Differential calculus provides the foundation for other math topics, allowing us to solve real-world problems by examining rates of change.

Function Derivatives

Derivatives are fundamental in calculus because they describe how a function behaves and changes over time. Let's break down the derivatives using the quotient rule, a key concept for finding the derivative of a function that is the ratio of two other functions.
The quotient rule is expressed as:\[ y' = \frac{u'v - uv'}{v^2} \] where:- \( u \) is the numerator function,- \( v \) is the denominator function,- \( u' \) and \( v' \) are the respective derivatives.

• Be sure to differentiate the top (numerator) and the bottom (denominator) functions separately.
• Substitute these derivatives back into the quotient rule formula to find the complete derivative.
• Double-check the expression to ensure simplification.

With function derivatives, you can better understand complex relationships within various functions, which is crucial for problem-solving in many fields including physics and economics.

Calculus in Biology

Calculus isn't just confined to math classes; it plays a vital role in the biological sciences as well. In biology, calculus helps us model population growth, understand speed of enzyme reactions, and even track movement patterns of animals.
Differents types of derivatives in calculus, like the one calculated using the quotient rule, helps biologists to:

• Predict how populations might change under different conditions.
• Understand how quickly diseases may spread.
• Estimate rates of environmental change.

By using calculus, biologists can transform complex biological phenomena into manageable equations, leading to better predictions and insights.