Question

Use implicit differentiation to find\(\frac{d y}{d x}.\) $$6 x^{3}+7 y^{3}=13 x y$$

Short answer

Answer: The derivative of y with respect to x is \(\frac{dy}{dx} = \frac{13y - 18x^2}{21y^2 - 13x}\).

Step-by-step solution

Differentiate both sides of the equation with respect to x

To begin, we differentiate both sides of the given equation, \(6x^3+7y^3 = 13xy\), with respect to x. Remember that we do not know an explicit expression for y as a function of x. Therefore, we need to apply the chain rule when differentiating terms containing y. Differentiate term by term.

Apply the chain rule to terms with y

When differentiating terms containing y, apply the chain rule: \(\frac{d}{dx}(y^3) = \frac{d}{dx}(y^3) \cdot \frac{dy}{dx} = 3y^2 \frac{dy}{dx}\).

Write down the implicit derivative

Differentiating both sides with respect to x and applying the chain rule, we get: $$\frac{d}{dx}(6x^3) + \frac{d}{dx}(7y^3) = \frac{d}{dx}(13xy)$$ $$18x^2 + 21y^2 \frac{dy}{dx} = 13(y + x\frac{dy}{dx})$$

Solve the equation for \(\frac{dy}{dx}\)

Now, we want to isolate \(\frac{dy}{dx}\) and solve for it explicitly. We achieve this by collecting all terms with \(\frac{dy}{dx}\) on one side and moving the rest to the other side: $$21y^2 \frac{dy}{dx} - 13x\frac{dy}{dx} = 13y - 18x^2$$ Factor out \(\frac{dy}{dx}\) from the left side: $$\frac{dy}{dx}(21y^2 - 13x) = 13y - 18x^2$$ Finally, divide through by \((21y^2 - 13x)\) to isolate \(\frac{dy}{dx}\): $$\frac{dy}{dx} = \frac{13y - 18x^2}{21y^2 - 13x}$$

Final answer

The derivative of y with respect to x is: $$\frac{dy}{dx} = \frac{13y - 18x^2}{21y^2 - 13x}$$

Key concepts

Understanding the Chain Rule

The chain rule is an essential tool in calculus, especially when dealing with implicit differentiation. It allows us to differentiate composite functions. In our scenario, since we are trying to differentiate expressions involving both \(x\) and \(y\) without knowing \(y\) as an explicit function of \(x\), the chain rule becomes indispensable.

The essence of the chain rule is that it helps to find the derivative of a function that is nested within another function. Consider the expression \(y^3\) in our equation. We want to take its derivative with respect to \(x\). Here, the chain rule expresses this differentiation process as: - Differentiate \(y^3\) with respect to \(y\), giving \(3y^2\).- Multiply the result by \(\frac{dy}{dx}\), acknowledging \(y\)'s dependence on \(x\).

Thus, applying the chain rule results in \(\frac{d}{dx}(y^3) = 3y^2 \frac{dy}{dx}\). The chain rule is particularly useful because it accommodates changes in \(y\) as \(x\) changes, even when a precise relationship isn't explicitly defined.

The Role of Derivatives in Implicit Differentiation

Derivatives measure the rate at which a function changes. In the exercise, we're interested in finding the derivative of \(y\) with respect to \(x\) from the implicit equation \(6x^3 + 7y^3 = 13xy\).

Since this involves expressions with both \(x\) and \(y\), and \(y\) is not isolated, we use implicit differentiation. It involves differentiating each term in the equation while applying rules, like the chain rule, to account for dependent relationships.

A key point to remember is that each time you differentiate \(y\) with respect to \(x\), you multiply by \(\frac{dy}{dx}\). For the term \(7y^3\):

• Differentiate \(7y^3\) to get \(21y^2\).
• Remember to multiply this by \(\frac{dy}{dx}\), writing it as \(21y^2\frac{dy}{dx}\).

This process captures the change in \(y\) with respect to \(x\) even when they aren't expressed as a direct function. Once the derivatives on both sides are calculated, you'll isolate \(\frac{dy}{dx}\) to solve for the rate of change.

Functions and Their Implicit Relationships

Functions define specific relationships between two quantities. In explicit functions, one quantity is presented directly in terms of another. However, not all relationships are neat and can be difficult to express explicitly. This is where implicit functions come in handy.

Take our equation: \(6x^3 + 7y^3 = 13xy\). This equation implicitly relates \(x\) and \(y\), but \(y\) is not presented directly with respect to \(x\). We must discern their relationship using implicit differentiation.

Implicit functions are useful when a dependent variable cannot be separated from the independent variable conveniently or when both variables are interdependent in a messier equation. They enable us to analyze the behavior of one variable relative to another without solving for one in terms of the other.

Understanding these concepts allows us to find how each variable in such relationships affects the other, which is the crux of solving problems involving implicit differentiation.