Problem 11 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, find \(d y / d x\) $$ \frac{x+y}{2 x-y}=1 $$

Step-by-Step Solution

Answer

The derivative of y with respect to x, \( \frac{dy}{dx}\), is the solution to this exercise. The specific value can be found by evaluating the derivative at any given value of x.

Step 1: Arrange the Equation

Rearrange the equation to leave the fraction on one side: \( \frac{x + y}{2x - y} - 1 = 0 \).

Step 2: Differentiation using the Quotient Rule

Use the Quotient Rule, which states that the derivative of \(\frac{u}{v}\) is \(\frac{vu' - uv'}{v^2}\), where u' and v' are the derivatives of u and v respectively. Here, u = x + y and v = 2x - y. Applying the rule gives: \( \frac{d y}{d x} = \frac{d}{d x} [\frac{x + y}{2x - y} - 1] \).

Step 3: Simplify the Derivative

Simplify the result obtained from Step 2 to find the derivative of y with respect to x. This is the final answer.

Key Concepts

Derivative

Understanding what a derivative in calculus means is foundational to grasping more complex concepts.
Think of a derivative as a way to measure how a quantity changes in response to changes in another quantity. For example, if you're driving a car, the derivative of the position of your car with respect to time would be the speed—the rate at which your car's position changes over time.

• The derivative of a function at a point calculates the rate at which the function’s value changes at that point.
• It is often denoted by \( f'(x) \) or \( \frac{df}{dx} \), where \( f \) signifies a function and \( x \) is the independent variable.
• When you find a derivative, you're calculating the slope of the function’s graph at any point, which corresponds to the steepness of the curve.

Derivatives have vast applications in multiple fields such as physics, economics, and biology. They help in modeling the real-world scenarios where one quantity changes with respect to another.

Quotient Rule

The quotient rule is one of the techniques used in calculus to find the derivative of a function that is the ratio of two differentiable functions.

• The quotient rule states that the derivative \( \frac{d}{dx} \left( \frac{u}{v} \right) \) of the function \( \frac{u}{v} \) where \( u \) and \( v \) are both functions of \( x \), is given by \( \frac{vu' - uv'}{v^2} \).
• Here, \( u' \) and \( v' \) represent the derivatives of \( u \) and \( v \) with respect to \( x \), respectively.
• This rule is particularly helpful when dividing one quantity by another, as it helps us understand how the ratio changes in response to changes in \( x \).

Applying the quotient rule requires careful attention to differentiating the numerator and the denominator and then subtracting and dividing according to the rule.

Calculus

Calculus is an extensive branch of mathematics focused on change and motion.
It's divided mainly into two areas:

• Differential Calculus: This deals with the derivatives and slopes of curves.
• Integral Calculus: This involves integration, which is essentially the reverse of differentiation and is concerned with areas under curves and accumulation of quantities.

Calculus is the mathematical language that describes and analyzes change, allowing us to create models for real-life situations like understanding the motion of planets or optimizing business processes. Mastery of calculus is essential for many scientific and engineering disciplines because it enables precise predictions and deep understanding of complex systems.

Differentiation

Differentiation is the action of computing a derivative.
It is the process by which we find the rate at which one quantity changes with respect to another. Differentiation helps us determine the exact rate of change at any point on a curve, which is a powerful tool in analyzing real-world problems.

• When we differentiate a function, we're looking for its instantaneous rate of change, much like an instant snapshot of how fast or slow it is moving at that specific point.
• There are various rules and techniques to differentiate functions, such as the power rule, product rule, chain rule, and the aforementioned quotient rule.
• Implicit differentiation, a technique used when dealing with functions not easily solvable for one variable, is when we differentiate both sides of an equation with respect to the independent variable.

Through differentiation, we can also determine the maximum and minimum points of curves, which is essential in optimization problems.