Problem 300 - Calculus Volume 1

Question

Use implicit differentiation to find \(\frac{d y}{d x}\). $$ x^{2}-y^{2}=4 $$

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = \frac{x}{y}\).

Step 1: Differentiate Both Sides of the Equation

We start by differentiating both sides of the equation with respect to \(x\). The given equation is \(x^{2} - y^{2} = 4\). When differentiating, remember to apply the derivative rules to each term separately. Differentiating the left side gives: \[ \frac{d}{dx}(x^2) - \frac{d}{dx}(y^2) = 0 \]

Step 2: Differentiate Each Term

Now focus on differentiating each term:- The derivative of \(x^2\) with respect to \(x\) is \(2x\).- The derivative of \(y^2\) with respect to \(x\) is \(2y \cdot \frac{dy}{dx}\) (using the chain rule).So, we have: \[ 2x - 2y \cdot \frac{dy}{dx} = 0 \]

Step 3: Solve for \(\frac{dy}{dx}\)

To isolate \(\frac{dy}{dx}\), we solve the equation from Step 2:\[ 2x - 2y \cdot \frac{dy}{dx} = 0 \]First, add \(2y \cdot \frac{dy}{dx}\) to both sides to get:\[ 2x = 2y \cdot \frac{dy}{dx} \]Next, divide both sides by \(2y\):\[ \frac{dy}{dx} = \frac{2x}{2y} \]Simplify to get:\[ \frac{dy}{dx} = \frac{x}{y} \]

Step 4: Conclusion Step: Interpret the Result

We have found that the derivative of \(y\) with respect to \(x\) is \(\frac{dy}{dx} = \frac{x}{y}\). This means that for any point \((x, y)\) on the curve defined by \(x^2 - y^2 = 4\), the slope of the tangent at that point is \(\frac{x}{y}\).

Key Concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus that is used to differentiate composite functions. You encounter this rule when dealing with situations where a function is nested inside another. The rule states that the derivative of a composite function is the derivative of the outer function multiplied by the derivative of the inner function.
To put this into more familiar terms, suppose you have a function \( f(g(x)) \). The Chain Rule tells us that the derivative, written as \( \frac{d}{dx}[f(g(x))] \), is \( f'(g(x)) \cdot g'(x) \). This method is especially useful when differentiating implicit functions, such as finding \( \frac{dy}{dx} \) for equations where \( y \) cannot be easily isolated.
The key to mastering the Chain Rule is practice. You'll frequently apply this rule to problems involving implicit differentiation or when encountering complex derivatives, such as the derivative of \( y^2 \) in the exercise, which requires using the chain rule to get \( 2y \cdot \frac{dy}{dx} \).

• Identify the outer and inner functions.
• Apply the Chain Rule by differentiating the outer function first.
• Multiply by the derivative of the inner function.

Derivative of Implicit Functions

In calculus, implicit differentiation is used when you have an equation involving both \( x \) and \( y \), and solving explicitly for \( y \) is difficult or impossible. Instead of solving for \( y \), you differentiate each term of the equation with respect to \( x \).
For the exercise equation \( x^2 - y^2 = 4 \), you differentiate directly as a whole. This involves differentiating \( x^2 \) and \( y^2 \) separately and applying the chain rule when differentiating \( y^2 \). Consequently, this yields the expression \( 2x - 2y \cdot \frac{dy}{dx} \).
Implicit differentiation allows us to find \( \frac{dy}{dx} \), which is the rate of change of \( y \) with respect to \( x \) without needing an explicit function definition for \( y \). By rearranging and solving these derivatives algebraically, we can express \( \frac{dy}{dx} \) in terms of both \( x \) and \( y \).

• Differentiate every term with respect to \( x \).
• Use the Chain Rule when necessary.
• Solve the resulting equation for \( \frac{dy}{dx} \).

Slope of Tangent Line

The slope of the tangent line to a curve at a given point provides valuable insights into the behavior and nature of the curve at that point. In mathematics, this slope is expressed as the derivative \( \frac{dy}{dx} \).
For the specific equation \( x^2 - y^2 = 4 \), after using implicit differentiation, the resulting expression for the derivative is \( \frac{dy}{dx} = \frac{x}{y} \). This derivative tells us how the curve is changing at any point \((x, y)\) on the graph.
The concept of the tangent line's slope is crucial because it helps to determine whether the curve is increasing or decreasing at a particular point. In cartesian coordinates, the tangent line at point \((a, b)\) has a slope \( m \) equal to \( \frac{a}{b} \), given by the derivative.

• The slope \( \frac{dy}{dx} \) shows the rate of change.
• Use this slope to determine the tangent line's equation.
• The slope \( \frac{x}{y} \) changes as you move along the curve.