Problem 26 - Thomas Calculus

Question

Use implicit differentiation to find \(d y / d x\) in Exercises \(19-32\) $$ x^{2}=\frac{x-y}{x+y} $$

Step-by-Step Solution

Answer

\( \frac{dy}{dx} = \frac{y - x(x+y)^2}{x} \)

Step 1: Differentiate both sides implicitly with respect to x

To solve the problem, we need to differentiate both sides of the equation with respect to x. The equation is: \[ x^2 = \frac{x-y}{x+y} \] By implicit differentiation: - Differentiate the left side with respect to x: \[ \frac{d}{dx}(x^2) = 2x \] - For the right side, we'll use the quotient rule for \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \). Let \(u = x - y \) and \(v = x + y\). So, \( \frac{d}{dx}(u) = 1 - \frac{dy}{dx} \) and \( \frac{d}{dx}(v) = 1 + \frac{dy}{dx} \).

Step 2: Apply the quotient rule to the right side

Applying the quotient rule to differentiate the right side:\[\frac{d}{dx}\left(\frac{x-y}{x+y}\right) = \frac{(x+y)(1 - \frac{dy}{dx}) - (x-y)(1 + \frac{dy}{dx})}{(x+y)^2}\]Let's expand and simplify:\[= \frac{(x+y) - (x+y)\frac{dy}{dx} - (x-y) - (x-y)\frac{dy}{dx}}{(x+y)^2}\]\[= \frac{x+y-x+y-(x+y)\frac{dy}{dx}+(x-y)\frac{dy}{dx}}{(x+y)^2}\]\[= \frac{2y - 2x\frac{dy}{dx}}{(x+y)^2}\]

Step 3: Set the derivatives equal and solve for \(\frac{dy}{dx}\)

Now, set the derivative from Step 1 equal to the result from Step 2:\[ 2x = \frac{2y - 2x\frac{dy}{dx}}{(x+y)^2} \]Multiply both sides by \((x+y)^2\) to clear the fraction:\[ 2x(x+y)^2 = 2y - 2x\frac{dy}{dx} \]Rearrange to solve for \(\frac{dy}{dx}\):\[ 2x\frac{dy}{dx} = 2y - 2x(x+y)^2 \]Divide by \(2x\) to isolate \(\frac{dy}{dx}\):\[ \frac{dy}{dx} = \frac{2y - 2x(x+y)^2}{2x} \]

Step 4: Simplify the result

Simplify the expression further:\[ \frac{dy}{dx} = \frac{y - x(x+y)^2}{x} \]

Key Concepts

Quotient Rule

The quotient rule is a special technique in calculus used to differentiate quotients of two functions. If you have a function that is a division of two other functions, the quotient rule provides a method to find its derivative. The rule states that for a function given as \(\frac{u}{v}\), its derivative with respect to \(x\) is:

• \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}\)

To apply the quotient rule, you need to:

• Identify \(u\) and \(v\), the numerator and denominator respectively.
• Differentiate \(u\) and \(v\) separately to find \(\frac{du}{dx}\) and \(\frac{dv}{dx}\).
• Plug these derivatives into the quotient rule formula.

By following these steps, you can find the derivative of any rational function effectively. It is particularly useful when you need to work with complex fractions in functions.

Differential Calculus

Differential calculus is a major branch of calculus that focuses on the concept of the derivative. It is fundamentally concerned with understanding how functions change and deals with the rate of change and slopes of curves. In essence, it is about finding out how a function behaves at any given point.

• The derivative of a function tells us the rate at which one quantity changes relative to another.
• Through differentiation, we can determine the slope of a function at any point, helping to understand its behavior better.
• Differential calculus helps in solving real-world problems related to motion, optimization, and rates of change in various physical systems.

The ability to differentiate functions allows mathematicians and scientists to model physical situations accurately by examining tangents and rates of change.

Derivative of Implicit Functions

When it comes to implicit functions, differentiation can be a bit tricky. Implicit differentiation is used when a relationship between variables is given in a non-explicit form. In other words, instead of \(y = f(x)\), you have an equation involving both \(x\) and \(y\) simultaneously, like \(x^2 = \frac{x-y}{x+y}\).
To differentiate such equations:

• Differentiating both sides of the equation with respect to \(x\) is the crucial step.
• Whilst differentiating, apply rules like the product rule, chain rule, or quotient rule as needed.
• Remember, every time you differentiate a \(y\) term, multiply by \(\frac{dy}{dx}\) because \(y\) is a function of \(x\).

By systematically applying rules of differentiation to every term and then solving for \(\frac{dy}{dx}\), you can find the derivative even when you cannot solve for \(y\) directly. Implicit differentiation is a powerful tool especially suited for complex equations where explicitly solving for one variable in terms of others isn't straightforward.