Math Problem Statement

Find d/dx (y) if y ^ 2 - x ^ 2 = 2

Solution

Given the equation:

y2x2=2y^2 - x^2 = 2

We need to find ddx(y)\frac{d}{dx}(y), or dydx\frac{dy}{dx}, using implicit differentiation. Let's proceed step by step:

Step 1: Differentiate both sides with respect to xx.

Differentiate the left-hand side term by term.

ddx(y2x2)=ddx(2)\frac{d}{dx}(y^2 - x^2) = \frac{d}{dx}(2)

Using the chain rule:

  • For y2y^2, treat yy as a function of xx (since y=y(x)y = y(x)):

ddx(y2)=2ydydx\frac{d}{dx}(y^2) = 2y \cdot \frac{dy}{dx}

  • For x2x^2:

ddx(x2)=2x\frac{d}{dx}(x^2) = 2x

On the right-hand side, the derivative of the constant 22 is 0:

ddx(2)=0\frac{d}{dx}(2) = 0

Step 2: Set up the equation.

After differentiating, we get:

2ydydx2x=02y \cdot \frac{dy}{dx} - 2x = 0

Step 3: Solve for dydx\frac{dy}{dx}.

First, isolate the term with dydx\frac{dy}{dx}:

2ydydx=2x2y \cdot \frac{dy}{dx} = 2x

Now, divide both sides by 2y2y:

dydx=xy\frac{dy}{dx} = \frac{x}{y}

Thus, the derivative dydx\frac{dy}{dx} is:

dydx=xy\frac{dy}{dx} = \frac{x}{y}


Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:

  1. How would you find d2ydx2\frac{d^2y}{dx^2} (the second derivative)?
  2. What happens if you substitute specific values for xx and yy?
  3. How would you interpret the result dydx=xy\frac{dy}{dx} = \frac{x}{y} graphically?
  4. Can you use implicit differentiation if the equation involves higher powers of yy?
  5. What is the physical interpretation of dydx=xy\frac{dy}{dx} = \frac{x}{y} in real-world scenarios?

Tip: Implicit differentiation is useful when it’s difficult or impossible to solve for one variable explicitly before differentiating.

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Math Problem Analysis

Mathematical Concepts

Implicit Differentiation
Derivatives
Chain Rule

Formulas

d/dx(y^2) = 2y * dy/dx
d/dx(x^2) = 2x

Theorems

Chain Rule of Differentiation

Suitable Grade Level

Grades 10-12