Problem 5 - The Calculus with Analytic Geometry

Question

Find \(D_{x} y\) by implicit differentiation. $$ \frac{1}{x}+\frac{1}{y}=1 $$

Step-by-Step Solution

Answer

Implicit differentiation gives \( \frac{dy}{dx} = \frac{y^2}{x^2} \).

Step 1 - Differentiate both sides with respect to x

Differentiate both sides of the equation \( \frac{1}{x}+\frac{1}{y}=1 \) with respect to x. Remember that y is a function of x, i.e., y = y(x). Use the chain rule for \( \frac{1}{y} \): \( \frac{d}{dx} \frac{1}{x} + \frac{d}{dx} \frac{1}{y} = \frac{d}{dx}(1) \)

Step 2 - Apply the derivative rules

Apply the derivative rules: \( \frac{d}{dx} \frac{1}{x} = -\frac{1}{x^2} \) and \( \frac{d}{dx} \frac{1}{y} = -\frac{1}{y^2} \frac{dy}{dx} \). Now the equation becomes: \[ -\frac{1}{x^2} + (-\frac{1}{y^2} \frac{dy}{dx}) = 0 \]

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

To isolate \( \frac{dy}{dx} \), add \( \frac{1}{x^2} \) to both sides: \[ -\frac{1}{y^2} \frac{dy}{dx} = \frac{1}{x^2} \] Now, multiply both sides by \( -y^2 \) to solve for \( \frac{dy}{dx} \): \[ \frac{dy}{dx} = \frac{y^2}{x^2} \]

Key Concepts

Chain Rule

The chain rule is an essential tool in calculus. It helps us differentiate complex functions. Whenever you differentiate a composite function, you use the chain rule. For example, if you have a function like \( \frac{1}{y} \), and you know \( y \) is actually a function of \( x \) (written as \( y(x) \)), you must apply the chain rule to find the derivative.

Here’s how the chain rule works in this exercise: Let's consider \( \frac{1}{y} \). When differentiating \( \frac{1}{y} \) with respect to \( x \), we treat \( y \) as a function of \( x \). The chain rule then states that:
\[ \frac{d}{dx} \frac{1}{y} = \frac{d}{dy} \frac{1}{y} \times \frac{dy}{dx} = -\frac{1}{y^2} \times \frac{dy}{dx} \]
This means we take the derivative of \( \frac{1}{y} \) with respect to \( y \) and then multiply by \( \frac{dy}{dx} \). Understanding this rule helps you handle implicit differentiation smoothly.

Derivative Rules

Using derivative rules is crucial in solving calculus problems. There are rules for differentiating various functions, and knowing them is key.

In our given equation \( \frac{1}{x} + \frac{1}{y} = 1 \), we first need to differentiate both sides with respect to \( x \).

This calls for applying the power rule and the chain rule:
1. The power rule shows us how to differentiate \( \frac{1}{x} \):
\[ \frac{d}{dx} \frac{1}{x} = -\frac{1}{x^2} \]
2. For \( \frac{1}{y} \), we apply the chain rule as discussed earlier. The derivative is:
\[ \frac{d}{dx} \frac{1}{y} = -\frac{1}{y^2} \frac{dy}{dx} \]

With this knowledge, we can rewrite our differentiated equation:
\[ -\frac{1}{x^2} + \frac{dy}{dx} \times -\frac{1}{y^2} = 0 \]
Mastering these rules and knowing when to apply them will make solving these types of problems easier.

Solving for dy/dx

The goal of implicit differentiation problems is often to solve for \( \frac{dy}{dx} \). This represents the rate of change of \( y \) with respect to \( x \).

After applying the chain rule and derivative rules, we rearrange the equation to isolate \( \frac{dy}{dx} \).
Here's the isolated form from our exercise:
\[ -\frac{1}{y^2} \frac{dy}{dx} = \frac{1}{x^2} \]
Next, to solve for \( \frac{dy}{dx} \), we need to eliminate the \( -\frac{1}{y^2} \) term from the left side. We can do this by multiplying both sides by \( -y^2 \). The equation then becomes:
\[ \frac{dy}{dx} = \frac{y^2}{x^2} \]
The solution shows how \( y \) changes with \( x \) by balancing both variables. Practice breaking it down step-by-step to build your confidence in solving for \( \frac{dy}{dx} \).