Question

Find \(\frac{d y}{d x}\) using implicit differentiation. $$\left(y^{2}+2 y-x\right)^{2}=200$$

Short answer

\(\frac{dy}{dx} = \frac{1}{2y + 2}\)

Step-by-step solution

Differentiate Both Sides

Differentiate both sides of the equation \( (y^2 + 2y - x)^2 = 200 \) with respect to \( x \). For the left side, apply the chain rule: the derivative of \( u^2 \) with respect to \( u \) is \( 2u \), where \( u = y^2 + 2y - x \).

Differentiate the Inside Function

Differentiate the inside function \( y^2 + 2y - x \) with respect to \( x \). This involves using the chain rule and implicit differentiation:\( \frac{d}{dx}[y^2] = 2y \frac{dy}{dx} \), \( \frac{d}{dx}[2y] = 2 \frac{dy}{dx} \), and \( \frac{d}{dx}[x] = 1 \).

Apply the Chain Rule

Apply the chain rule to combine the results. The derivative for the left side of the equation becomes:\[2(y^2 + 2y - x)(2y \frac{dy}{dx} + 2\frac{dy}{dx} - 1)\].

Set the Derivative of the Right Side

Since the right side of the equation is 200, its derivative is zero.

Combine and Simplify

Set the derivative of the left side equal to the derivative of the right side:\[2(y^2 + 2y - x)(2y \frac{dy}{dx} + 2\frac{dy}{dx} - 1) = 0\]This implies either \( y^2 + 2y - x = 0 \) or the expression \( 2y \frac{dy}{dx} + 2\frac{dy}{dx} - 1 = 0 \) must be zero.

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

If \( 2y \frac{dy}{dx} + 2\frac{dy}{dx} - 1 = 0 \), factor out \( \frac{dy}{dx} \):\[(2y + 2)\frac{dy}{dx} = 1\]Thus, solving for \( \frac{dy}{dx} \) gives:\[\frac{dy}{dx} = \frac{1}{2y + 2}\].

Key concepts

Chain Rule

The chain rule is a fundamental concept in calculus used to find the derivative of composite functions. It tells us how to differentiate a function whose variable itself is a function of another variable.

Imagine a function inside a function, like peeling an onion. When applying the chain rule:

• Identify the outer function and the inner function.
• Differentiate the outer function as if the inner function was just a simple variable.
• Multiply this by the derivative of the inner function.

In the exercise, we use the chain rule to differentiate the main expression y \( (y^2 + 2y - x)^2 \) since it has another function inside (\( y^2 + 2y - x \)). We start by differentiating the outer layer, \(u^2\), where \(u = y^2 + 2y - x\). Its derivative is \(2u\), and then, of course, follow up by working with the inner derivative for our full solution.

Derivative

Derivatives measure how a function changes as its input changes. Think of it as the "rate of change." For example, in physics, velocity is the derivative of position concerning time.

When dealing with derivatives, we calculate:

• The slope of a function at any point.
• How one quantity changes concerning another.

In mathematics, particularly calculus, we represent the derivative of \( y \) concerning \( x \) as \( \frac{d y}{d x} \). In our exercise, we explore the derivative of the equation \( (y^2 + 2y - x)^2 = 200 \), seeing how it behaves when both \( x \) and \( y \) are in complex interplay.Different techniques, such as implicit differentiation and the chain rule, help simplify how these derivatives emerge from intertwined variables or complex functions.

Implicit Functions

Implicit functions arise when you cannot easily solve for one variable in terms of another. Instead of \( y = f(x) \), we deal with an expression like \( F(x, y) = 0 \). This requires a different differentiation approach.

Implicit differentiation steps in here because:

• Directly solving for \( y \) might be impossible or cumbersome.
• We can find derivatives without explicitly isolating the variable.

In our example, \( (y^2 + 2y - x)^2 = 200 \) is not easily solved for \( y \), but implicit differentiation allows us to find \( \frac{d y}{d x} \) nonetheless. Implicit differentiation treats \( y \) as a function of \( x \) and differentiates every term concerning \( x \), acknowledging the hidden dependency of \( y \) on \( x \). This approach leverages derivatives like \( \frac{d y}{d x} \) while respecting the interconnectedness of the terms involved.

Differentiation Steps

Differentiation often involves a systematic process, especially when tackling implicit equations. Let's talk about the differentiation steps relevant to our example:

1. **Differentiate Both Sides:** Start by applying the chain rule to the left side \( (y^2 + 2y - x)^2 = 200 \), differentiating \( y^2 + 2y - x \) first.2. **Differentiate the Inside Function:** Each segment within \( y^2 + 2y - x \) must be differentiated. Calculate individually: - \( \frac{d}{dx}[y^2] = 2y \frac{dy}{dx} \) - \( \frac{d}{dx}[2y] = 2 \frac{dy}{dx} \) - \( \frac{d}{dx}[x] = 1 \)3. **Apply the Chain Rule:** Combine the inner derivatives with \( 2u \) from the outer function: - \( 2(y^2 + 2y - x)(2y \frac{dy}{dx} + 2 \frac{dy}{dx} - 1) \)4. **Set the Derivative of the Right Side:** The derivative of a constant, 200, is zero.5. **Combine and Simplify:** Set both sides equal and simplify: - First option: \( y^2 + 2y - x = 0 \) - Second option and solution: Solve \( 2y \frac{dy}{dx} + 2 \frac{dy}{dx} - 1 = 0 \) which guides us to \( \frac{dy}{dx} = \frac{1}{2y + 2} \).Through these steps, we efficiently uncover how the variables interact and reveal the derivative we seek.