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Magazine Chapter 2: Problem 11
Find \(d y / d x\) by implicit differentiation. $$\tan (x / y)=x+y$$
Short Answer
Expert verified
\( \frac{dy}{dx} = \frac{y \sec^2(\frac{x}{y}) - y^2}{y^2 + x \sec^2(\frac{x}{y})} \)
Step by step solution
01
Differentiate Both Sides with Respect to x
To find \( \frac{dy}{dx} \) using implicit differentiation, start by differentiating both sides of the equation \( \tan(\frac{x}{y}) = x + y \) with respect to \( x \). Recall that the derivative of \( \tan(u) \) is \( \sec^2(u) \cdot \frac{du}{dx} \).
02
Apply Chain Rule on Left Side
For the left side, \( \tan(\frac{x}{y}) \), use the chain rule: the derivative of \( \tan(u) \) is \( \sec^2(u) \) and apply the chain rule to \( \frac{x}{y} \), giving \( \left(\frac{y \cdot 1 - x \cdot \frac{dy}{dx}}{y^2}\right) \). Thus, \( \sec^2(\frac{x}{y}) \cdot \frac{y - x \frac{dy}{dx}}{y^2} \).
03
Differentiate Right Side Terms
Differentiate \( x + y \) with respect to \( x \). The derivative is \( 1 + \frac{dy}{dx} \).
04
Set Derivatives Equal
Set the derivatives obtained from differentiating both sides equal: \[\sec^2(\frac{x}{y}) \cdot \frac{y - x \frac{dy}{dx}}{y^2} = 1 + \frac{dy}{dx}\]
05
Solve for \( \frac{dy}{dx} \)
Distribute across the left side and solve for \( \frac{dy}{dx} \). This involves clearing the fraction and combining like terms. The equation to solve is: \[ \frac{y \sec^2(\frac{x}{y})}{y^2} - \frac{x \sec^2(\frac{x}{y}) \frac{dy}{dx}}{y^2} = 1 + \frac{dy}{dx} \] Rearrange and solve for \( \frac{dy}{dx} \).
06
Final Simplification
Simplify the expression to isolate terms involving \( \frac{dy}{dx} \) to one side. After rearranging, we find: \[ \frac{dy}{dx} = \frac{y \sec^2(\frac{x}{y}) - y^2}{y^2 + x \sec^2(\frac{x}{y})} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding the Chain Rule
When tackling implicit differentiation, the chain rule is a key tool. The chain rule helps us determine the derivative of a composite function.
In our exercise, we use it to differentiate the tangent function, specifically the expression \( \tan(\frac{x}{y}) \).
The chain rule states that the derivative of \( f(g(x)) \) is the derivative of \( f \) with respect to \( g(x) \) times the derivative of \( g(x) \) with respect to \( x \).
In our exercise, we use it to differentiate the tangent function, specifically the expression \( \tan(\frac{x}{y}) \).
The chain rule states that the derivative of \( f(g(x)) \) is the derivative of \( f \) with respect to \( g(x) \) times the derivative of \( g(x) \) with respect to \( x \).
- Here, \( f(u) = \tan(u) \), and we know the derivative of \( \tan(u) \) is \( \sec^2(u) \).
- For \( g(x) = \frac{x}{y} \), its derivative involves both \( x \) and \( y \). Hence, it is \( \frac{y - x \frac{dy}{dx}}{y^2} \).
Derivative of the Tangent Function and Implicit Differentiation
The tangent function \( \tan(x) \) has a characteristic derivative, often represented as \( \sec^2(x) \).
In our given equation, we have \( \tan(\frac{x}{y}) = x + y \). Applying implicit differentiation means differentiating every part of the equation with respect to \( x \).
Since \( y \) is a function of \( x \), we use the chain rule when it appears in denominators or numerators.In the context of our exercise:
In our given equation, we have \( \tan(\frac{x}{y}) = x + y \). Applying implicit differentiation means differentiating every part of the equation with respect to \( x \).
Since \( y \) is a function of \( x \), we use the chain rule when it appears in denominators or numerators.In the context of our exercise:
- The left side, \( \tan(\frac{x}{y}) \), requires differentiating as \( \sec^2(\frac{x}{y}) \) using the chain rule for the innermost function.
- Combining the derivative of the inside expression \( \frac{x}{y} \), results in \( \sec^2(\frac{x}{y}) \times \frac{y - x \frac{dy}{dx}}{y^2} \).
Solving Differential Equations in Implicit Differentiation
Once both sides of the equation are differentiated, we equate these to solve for \( \frac{dy}{dx} \).
This is essentially solving a differential equation. The equation \( \sec^2(\frac{x}{y}) \cdot \frac{y - x \frac{dy}{dx}}{y^2} = 1 + \frac{dy}{dx} \) needs rearranging, combining like terms, and simplifying to isolate \( \frac{dy}{dx} \).Here's the strategy:
This is essentially solving a differential equation. The equation \( \sec^2(\frac{x}{y}) \cdot \frac{y - x \frac{dy}{dx}}{y^2} = 1 + \frac{dy}{dx} \) needs rearranging, combining like terms, and simplifying to isolate \( \frac{dy}{dx} \).Here's the strategy:
- Multiply through to remove fractions, ensuring all terms involving \( \frac{dy}{dx} \) are grouped together.
- Rearrange terms to get \( \frac{dy}{dx} \) on one side. This means moving terms with \( \frac{dy}{dx} \) and constants to solve for the variable.
- Simplify by factoring and reducing equations whenever possible.
