Problem 21 - AP Calculus with 8 Practice Tests 15th

Question

\(Y\) is a differentiable function of \(x\) Choose the alternative that is the derivative \(\frac{d y}{d x}\). \(x^{3}-y^{3}=1\) (A) \(x\) (B) \(3 x^{2}\) (C) \(\frac{x^{2}}{y^{2}}\) (D) \(\frac{3 x^{2}-1}{y^{2}}\)

Step-by-Step Solution

Answer

Choice (C) \(\frac{x^2}{y^2}\) is the derivative.

Step 1: Apply Implicit Differentiation

Start by differentiating the given equation \(x^3 - y^3 = 1\) with respect to \(x\). Since \(y\) is a function of \(x\), we will use implicit differentiation. Differentiate both sides: \(3x^2 - 3y^2 \frac{dy}{dx} = 0\).

Step 2: Solve for \(\frac{dy}{dx}\)

Rearrange the equation to isolate the term with \(\frac{dy}{dx}\): \(3y^2 \frac{dy}{dx} = 3x^2\).

Step 3: Simplify the Expression

Divide both sides of the equation by \(3y^2\) to solve for \(\frac{dy}{dx}\): \(\frac{dy}{dx} = \frac{3x^2}{3y^2}\). Simplify this expression: \(\frac{dy}{dx} = \frac{x^2}{y^2}\).

Step 4: Match with Multiple Choices

Compare the derived expression \(\frac{dy}{dx} = \frac{x^2}{y^2}\) to the given choices. The correct matching choice is (C) \(\frac{x^2}{y^2}\).

Key Concepts

derivative

In calculus, the derivative of a function measures how the output value of a function changes as the input value changes. Essentially, it tells us the rate of change or the slope of the function at any given point. In our exercise, we dealt with the equation \(x^{3} - y^{3} = 1\). The task was to find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).

To find this derivative, we used a technique called implicit differentiation, which is particularly useful when \(y\) is not isolated on one side of the equation, as it is when the function cannot be solved explicitly for \(y\).

By taking the derivative of each term in the equation with respect to \(x\), we derive a new equation involving \(\frac{dy}{dx}\). Knowing how to take derivatives is crucial for understanding rates of change in a variety of contexts, from physics to economics.

differentiable function

A differentiable function is one that has a derivative at every point in its domain. If a function is differentiable, it is also continuous. This concept is crucial in many calculus problems and ensures smooth changes over its domain.

In the problem given, it states that \(Y\) is a differentiable function of \(x\). This tells us that \(y\) will smoothly and predictably change with each change in \(x\). Thus, we can confidently use calculus tools, such as derivatives, to analyze and understand its behavior.

Understanding whether a function is differentiable helps identify if it is possible to find a derivative at every point, which is important when making predictions about the function's rate of change.

solve for dy/dx

To solve for \(\frac{dy}{dx}\), we typically start by differentiating the equation using implicit differentiation. Looking at our problem, after differentiating the equation \(x^{3} - y^{3} = 1\), we landed on \(3x^{2} - 3y^{2}\frac{dy}{dx} = 0\).

From here, the next step is to isolate \(\frac{dy}{dx}\). We do this by manipulating the algebraic equation to get \(\frac{dy}{dx}\) alone on one side of the equation. Through rearranging, we obtained \(3y^{2} \frac{dy}{dx} = 3x^{2}\).

Finally, dividing both sides by \(3y^{2}\), we get \(\frac{dy}{dx} = \frac{x^{2}}{y^{2}}\). This is the derivative of \(y\) with respect to \(x\), showing us how \(y\) changes as \(x\) varies. By solving for \(\frac{dy}{dx}\), we are able to determine the rate at which \(y\) changes in relation to \(x\), a key output of the implicit differentiation process.