Question

Find \(d y / d x\). (Treat \(a\) and \(r\) as constants.) $$x^{2}+y^{2}=r^{2}$$

Short answer

\(\frac{dy}{dx} = -\frac{x}{y}\)

Step-by-step solution

Express the equation in the implicit form

Start with the given equation of a circle: \(x^{2} + y^{2} = r^{2}\). This equation is already in the implicit form, which means both variables, x and y, are on the same side of the equation.

Differentiate implicitly with respect to x

To find \(\frac{dy}{dx}\), we need to differentiate both sides of the equation with respect to x. This gives us \(2x + 2y\frac{dy}{dx} = 0\). The term \(2y\frac{dy}{dx}\) comes from applying the chain rule to \(y^{2}\) as y is a function of x.

Solve for \(dy/dx\)

Isolate \(\frac{dy}{dx}\) by moving the term involving x to the other side, resulting in \(2y\frac{dy}{dx} = -2x\). Now, divide both sides of the equation by \(2y\) to get \(\frac{dy}{dx} = -\frac{x}{y}\).

Key concepts

Differential Calculus

Differential calculus is a subsection of calculus concerned with the study of how functions change when their inputs change. It's primarily focused on the concept of the derivative, which measures the rate at which a quantity changes. In the context of our exercise, we look at the rate of change of one variable, namely, the y-coordinate of a point, with respect to another, which is the x-coordinate.

To compute this rate of change, or derivative, for the equation of a circle, we need to employ techniques that allow us to handle equations involving both x and y without solving explicitly for one in terms of the other. This type of equation presents a typical scenario where differential calculus shines by providing tools like implicit differentiation to handle such cases.

Chain Rule

The chain rule is a fundamental rule in differential calculus used to calculate the derivative of a composition of functions. It's based on the idea that if a variable z depends on the variable y, which itself depends on the variable x, then the rate at which z changes with respect to x can be found by multiplying the rate at which z changes with respect to y by the rate at which y changes with respect to x.

During the implicit differentiation of the circle's equation, \(2x + 2y\frac{dy}{dx} = 0\), we use the chain rule to differentiate \(y^2\). Since \(y\) is a function of \(x\), we treat \(y^2\) as the outer function \(u^2\) with the inner function \(u = y(x)\). This allows us to first find the derivative of the outer function and then multiply it by the derivative of the inner function, \(\frac{dy}{dx}\), effectively 'chaining' them together.

Equations of a Circle

The equations of a circle are mathematical representations that define the set of all points in a plane that are at a fixed distance, known as the radius, from a given point, called the center. The standard form of the circle's equation is \(x^2 + y^2 = r^2\), where \(r\) is the radius, and the center is at the origin \(0,0\).

The equation encompasses all the points \(x,y\) that satisfy this condition of equidistance from the center. It's crucial for problems in geometry and calculus. Since the circle's equation involves two variables, we often have to use implicit differentiation to find the relationship between their derivatives when the equation does not allow or it's not convenient to isolate the variables.

Implicit Function Theorem

The implicit function theorem is a powerful and extensive concept in calculus. It states that if a function can be written in a form that relates two or more variables implicitly, then under certain conditions, these variables can be expressed as differentiable functions with respect to one another.

In layman's terms, it makes sure that under the right circumstances, we can treat equations like \(x^2 + y^2 = r^2\) as though there's an underlying function \(y=f(x)\) or \(x=g(y)\) being hidden from us. With this assurance, we can proceed confidently to differentiate implicitly without needing an explicit formula for \(y\) in terms of \(x\) or vice versa. Implicit differentiation, predicated on this theorem, facilitates calculating derivatives for equations that define y uniquely in terms of x, even when those relationships are not expressed directly.