Question

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$5 x-2 y=7$$

Short answer

The derivative of y with respect to x, dy/dx, is \frac{5}{2}.

Step-by-step solution

Implicit Differentiation

Differentiate both sides of the equation with respect to x. Remember that while differentiating y with respect to x, treat y as a function of x (i.e., y=y(x)) and apply the chain rule.

Apply the Derivatives

Differentiate 5x to get 5. Differentiate -2y with respect to x to get -2(dy/dx), because the derivative of y with respect to x is dy/dx.

Solve for dy/dx

Isolate dy/dx on one side of the equation to solve for the derivative of y with respect to x.

Key concepts

Chain Rule

The chain rule is a fundamental principle in calculus used to find the derivative of composite functions. When you have a function that is composed of other functions, such as y being a function of x, the chain rule allows you to differentiate it effectively.

Think of it as peeling an onion, layer by layer. If you want to differentiate a function that's 'inside' another function, you need to multiply the derivative of the outer function by the derivative of the inner function. In mathematical terms, if you have an equation with y = f(g(x)), the chain rule states that the derivative of y with respect to x is f'(g(x)) * g'(x).

In the context of implicit differentiation, like in our original exercise, we treat y as a function of x even if it's not explicitly stated. When we differentiate any term with y in it, we must apply the chain rule by also differentiating y with respect to x.

Derivative of y with respect to x

When working with equations involving two variables, x and y, and we are interested in the rate at which y changes with respect to x, we look for the derivative of y with respect to x, often expressed as dy/dx. This concept is vital in understanding the relationship between the variables in an equation.

To compute dy/dx, we use the rules of differentiation. If y is an explicit function of x—like y = x^2—it is straightforward. However, when y is not isolated, as in the exercise example 5x - 2y = 7, we must differentiate implicitly. This process involves treating y as a function of x (i.e., y(x)) and applying the appropriate differentiation rules to both sides of the equation. The goal is to rearrange the equation and isolate dy/dx to find its value.

Solving Differential Equations

Differential equations involve derivatives and are powerful tools in expressing how quantities change. Solving a differential equation means finding the function or functions that fulfill the equation.

In our textbook exercise, we have a simple form of differential equation where we are seeking the derivative dy/dx. The process to solve it involves three major steps: differentiation using implicit differentiation, isolating dy/dx, and simplifying the equation. Once we apply the chain rule and differentiate with respect to x, we aim to express dy/dx solely in terms of x (and any other constants present).

For more complex equations, solving a differential equation can involve integration, special functions, and even numerical methods. However, the goal remains to express the relationship between the changing quantities in a way that can be understood and used for further analysis or application.