Problem 39 Assume that $x$ and $y$ are ... [FREE SOLUTION]

Chapter 3: Problem 39

Assume that $x$ and $y$ are differentiable functions of $t$. Find $d y / d t$ in terms of $x, y$, and $d x / d t$. $$ y^{2}-x^{2}=4 $$

Short Answer

Expert verified

$ \frac{dy}{dt} = \frac{x}{y} \frac{dx}{dt} $

Step by step solution

Differentiate Both Sides

Differentiate both sides of the equation with respect to the variable $ t $. The equation given is $ y^2 - x^2 = 4 $. Taking the derivative of both sides, we get:\[ \frac{d}{dt}(y^2) - \frac{d}{dt}(x^2) = \frac{d}{dt}(4) \]

Apply the Chain Rule

Use the chain rule for differentiation. The chain rule states that $ \frac{d}{dt}(y^2) = 2y \frac{dy}{dt} $ and $ \frac{d}{dt}(x^2) = 2x \frac{dx}{dt} $. So our equation becomes:\[ 2y \frac{dy}{dt} - 2x \frac{dx}{dt} = 0 \]

Solve for $ \frac{dy}{dt} $

We need to isolate $ \frac{dy}{dt} $ in the equation from Step 2. To do so, start by adding $ 2x \frac{dx}{dt} $ to both sides of the equation:\[ 2y \frac{dy}{dt} = 2x \frac{dx}{dt} \]Next, divide both sides by $ 2y $ to solve for $ \frac{dy}{dt} $:\[ \frac{dy}{dt} = \frac{x}{y} \frac{dx}{dt} \]

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Key Concepts

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

Chain Rule

The chain rule is a fundamental concept in calculus used for differentiating composite functions. Think of it as peeling layers of an onion; you work from the outside in, differentiating each layer as you go.
In this exercise, we start by considering the derivative of expressions like $ y^2 $ and $ x^2 $ with respect to $ t $. These expressions involve functions $ y(t) $ and $ x(t) $, which are composed of another function $ t $.

The chain rule tells us how to differentiate these by expressing the dependency on $ t $. For $ y^2 $, we use it to find $ \frac{d}{dt}(y^2) = 2y \frac{dy}{dt} $. This indicates how $ y^2 $ changes with respect to $ t $, involving both the value of $ y $ and its rate of change over time $ \frac{dy}{dt} $.
Similarly, for $ x^2 $, apply the chain rule: $ \frac{d}{dt}(x^2) = 2x \frac{dx}{dt} $. Here, it ties together the current state of $ x $ and its rate of change, $ \frac{dx}{dt} $.

This powerful tool allows us to handle more complex relationships between variables that are functions of another variable, like time here.

Implicit Differentiation

Implicit differentiation is another critical technique when working with functions that are not easily solved for one variable in terms of another. Functions like $ y^2 - x^2 = 4 $ are expressed implicitly rather than explicitly.
In our exercise, instead of solving explicitly for $ y $ or $ x $, we differentiate both sides with respect to $ t $.

When you see $ y^2 $ and $ x^2 $, understand that these are implicit functions of $ t $. We apply the differentiation rules (using the chain rule from before) directly to these expressions.
The process maintains the relationship between $ x $, $ y $, and their respective rates of change with respect to $ t $.

By differentiating implicitly, we keep the integrity of the equation while deriving useful information about how one set of changes fuels another, aligning it beautifully for solving problems involving related rates.

Related Rates

Related rates problems involve finding the rate at which one quantity changes with respect to another. This naturally extends from differential equations like $ y^2 - x^2 = 4 $.
The aim here is to find the relationship between the changes in $ y $ and $ x $ as they relate to time, $ t $.

In this exercise, once we differentiate the equation, both the chain rule and implicit differentiation reveal the differential equation: $ 2y \frac{dy}{dt} - 2x \frac{dx}{dt} = 0 $.
Solving for $ \frac{dy}{dt} $ gives us a formula: $ \frac{dy}{dt} = \frac{x}{y} \frac{dx}{dt} $. This equation says if you know how $ x $ changes over time, you can calculate $ \frac{dy}{dt} $ based on $ x $ and $ y $'s current values.

Related rates often appear in physical applications where multiple quantities depend on time. It's a means to intersect the geometry of change with algebraic relationships, offering predictive insights about systems.

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91影视

Assume that \(x\) and \(y\) are differentiable functions of \(t\). Find \(d y / d t\) in terms of \(x, y\), and \(d x / d t\). $$ y^{2}-x^{2}=4 $$

Short Answer

Step by step solution

Differentiate Both Sides

Apply the Chain Rule

Solve for \( \frac{dy}{dt} \)

Key Concepts

Chain Rule

Implicit Differentiation

Related Rates

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