Question

In Exercises, find \(d y / d x\) $$ x y=4 $$

Short answer

The derivative \(dy / dx\) of the equation \(xy = 4\) is \(-y / x\).

Step-by-step solution

Identify the variables in the equation

In our case, we have two variables which are x and y. The given equation is \(xy = 4\).

Differentiate both sides of the equation

Applying the product rule, the derivative of \(x\) with respect to \(x\) times \(y\) plus \(x\) times the derivative of \(y\) with respect to \(x\) equals the derivative of the constant \(4\) with respect to \(x\). This results in \(1 \cdot y + x \cdot dy/dx = 0\).

Solve for \(dy/dx\)

We isolate \(dy/dx\) to one side of the equation to find its value. Subtract \(1 \cdot y\) from both sides and then divide the whole equation by \(x\). We get \(dy/dx = -y / x\).

Key concepts

Product Rule

In differential calculus, the product rule is a fundamental technique used to find the derivative of products of two or more functions. It states that if you have a product of functions, let's say u(x) and v(x), the derivative of their product with respect to x is given by u'v + uv', where u' and v' represent the derivatives of u and v respectively.

For instance, if we have an equation such as xy = 4, and we wish to differentiate with respect to x, we can apply the product rule. Label x as our first function (which we'll call u) and y as the second function (which we'll call v). Derivatives of u and v are 1 and dy/dx respectively (since y is implicitly a function of x). Using the product rule formula, we find that the derivative of xy with respect to x is 1 * y + x * (dy/dx).

Understanding and correctly applying the product rule is essential when dealing with products of variable terms where each is a function of the differential variable.

Differential Calculus

Differential calculus is a branch of mathematics that studies the rates at which quantities change. This field is fundamentally concerned with the concept of a derivative, which provides an exact mathematical way to analyze and calculate this rate of change.

In practical terms, when we face a function f(x), the derivative of this function, denoted as f'(x) or df/dx, tells us how f(x) changes as x varies. Taking our example of xy = 4, this equation can represent a curve where every point on the curve satisfies the equation. By using differential calculus, specifically implicit differentiation in this case, we can understand how y changes in response to changes in x without the need to solve for y explicitly. This is particularly useful for complex equations or when y cannot be easily expressed as a function of x.

Derivative of Constants

In calculus, constants play a special role in differentiation. A constant is a term that does not depend on the variable of differentiation. As such, the derivative of a constant with respect to any variable is zero. This concept is essential to understand and apply when performing differentiation in various contexts.

Referring to our equation xy = 4, the number 4 on the right-hand side of the equation is a constant. When differentiating any constant like this with respect to x, the result is 0. This is why in the solution provided for the example, when we implicitly differentiate both sides of the equation with respect to x, the right-hand side becomes 0 after differentiation. This principle greatly simplifies many differentiation processes, as it allows us to dismiss any constant terms and focus on the differentiation of variable-dependent terms.