Question

Write the differential \(d y\) in terms of \(x, y,\) and \(d x\) for each implicit relation. $$2 \sqrt{x}+3 \sqrt{y}=4$$

Short answer

Using implicit differentiation, the differential dy in terms of x, y, and dx is dy = -\frac{\sqrt{x}}{\sqrt{y}} dx.

Step-by-step solution

Differentiate both sides of the equation with respect to x

To find the differential dy in terms of x, y, and dx, take the derivative of both sides of the equation with respect to x, treating y as a function of x (since the relationship between x and y is implicit). Remember to use the chain rule for differentiating the square root of y.

Solve the derivative for dy

After differentiating, isolate the term involving dy on one side of the equation to solve for dy in terms of dx and the other variables x and y.

Simplify the expression

Simplify the expression you obtained in Step 2 to get dy expressed only in terms of x, y, and dx.

Key concepts

Differential Calculus

Differential calculus is a branch of mathematics that deals with the study of rates at which quantities change. It's fundamental for understanding motion, change in systems, and the concept of instantaneous rate of change. One of its primary tools is the derivative, which measures how a function's output value changes in response to a change in its input value.

When faced with an equation involving two variables, like the one in the exercise \( 2 \sqrt{x} + 3 \sqrt{y} = 4 \), differential calculus enables us to determine the relationship between small changes in these variables—denoted by \(dx\) and \(dy\). The operation of finding \(dy\) in terms of \(dx\), \(x\), and \(y\) is called implicit differentiation, which is a technique used when a function is not given explicitly as \(y=f(x)\) but is instead embedded within an equation.

Chain Rule

The chain rule is an essential principle in calculus used to calculate the derivative of composite functions. When dealing with functions where one quantity depends on another, which in turn depends on a third variable, the chain rule provides a method to differentiate these interrelated components.

In the context of the original exercise, we have a term \(3\sqrt{y}\) that needs to be differentiated with respect to \(x\). Here, \(y\) is a function of \(x\), but it is nested inside a square root. To differentiate this term correctly, we apply the chain rule: first, we differentiate the outer function (the square root in this case), and then multiply by the derivative of the inner function (\(y\) with respect to \(x\)). This process reveals the presence of \(dy/dx\) within our expression, which is key to solving for \(dy\) in terms of \(dx\).

Solving Derivatives

Solving derivatives involves finding the expression for the rate of change of one variable relative to another. To solve a derivative, you typically perform differentiation on a function. However, when dealing with an implicit relationship, such as an equation where \(y\) is not expressed explicitly as a function of \(x\), the process involves a few extra steps, since \(y\) itself changes as \(x\) changes.

After applying the chain rule as we did in the previous section, you'll get an equation that contains \(dx\), \(dy\), and derivatives of \(y\) with respect to \(x\). The next task is to isolate \(dy\) to express it in terms of \(dx\), \(x\), and \(y\). This may require manipulating algebraic expressions and combining like terms. Once you isolate \(dy\), you have solved for the derivative in terms of the original variables and can understand how a small change in \(x\) can lead to a change in \(y\).