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Chapter 3: Problem 9

Suppose that \(x\) and \(y\) are related by the given equation and use implicit differentiation to determine \(\frac{d y}{d x}\). $$x y=5$$

Short Answer

Expert verified
The implicit derivative \( \frac{d y}{dx} = -\frac{y}{x} \).

Step by step solution

01

- Differentiate both sides with respect to x

Differentiate the given equation implicitly. Remember that y is a function of x, so use the product rule when differentiating the left-hand side. Start with \( x y = 5 \), which gives \( \frac{d}{dx}(xy) = \frac{d}{dx}(5) \). Since 5 is a constant, \( \frac{d}{dx}(5) = 0 \).
02

- Apply the product rule

Apply the product rule to the left-hand side \( \frac{d}{dx}(xy) \), which states that \( \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \). Here, \( u = x \) and \( v = y \), so \( \frac{d}{dx}(xy) = x \frac{d y}{dx} + y \).
03

- Set up the differentiated equation

Combine the results from Steps 1 and 2 to get the differentiated equation: \( x \frac{d y}{dx} + y = 0 \).
04

- Solve for \( \frac{d y}{dx} \)

Isolate \( \frac{d y}{dx} \) by first subtracting \( y \) from both sides: \( x \frac{d y}{dx} = -y \), then divide both sides by \( x \) to obtain \( \frac{d y}{dx} = -\frac{y}{x} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

product rule
When differentiating functions that are multiplied together, the product rule is a must-know trick. The product rule states that if you have two functions, say \(u\) and \(v\), then the derivative of their product \( uv \) with respect to \( x \) is given by:\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]This rule helps break down the derivative into manageable parts. In our exercise, \(u\) is \( x \) and \(v\) is \( y \). Since \( y \) is a function of \( x \), we need to use the product rule to correctly differentiate \( xy \). Applying the product rule, we get \( x \frac{d y}{dx} + y \).
derivatives
A derivative represents the rate at which a function changes. In simple terms, it tells us how a function's output value changes as its input value changes. In the context of the exercise, we need to find the derivative of both sides of the equation \( x y = 5 \):
  • Differentiate the left side using the product rule.
  • Differentiate the right side, which is a constant, so its derivative is \( 0 \).
This effectively gives us the rate at which \( y \) changes as \( x \) changes. It's important to carry out the derivatives correctly to solve for \( \frac{d y}{dx} \).
calculating dy/dx
To determine \( \frac{d y}{dx} \) using implicit differentiation, follow these steps:
  • Start from your given equation and perform implicit differentiation on both sides.
  • Apply the product rule to the left-hand side to handle the multiplication of \( x \) and \( y \).
  • Simplify the resulting expression to isolate \( \frac{d y}{dx} \).
For our example, we have:
1. Start with \( x y = 5 \).
2. Differentiate implicitly: \( \frac{d}{dx}(xy) = \frac{d}{dx}(5) \).
3. Apply the product rule: \( x \frac{d y}{dx} + y \).
4. The right side is a constant, so it differentiates to \( 0 \).
5. We get: \( x \frac{d y}{dx} + y = 0 \).
Next, solve for \( \frac{d y}{dx} \): subtract \( y \) from both sides and divide by \( x \): \[ \frac{d y}{dx} = - \frac{y}{x} \]Now, you have successfully calculated the derivative \( \frac{d y}{dx} \).

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