Math Problem Statement

Solve the problem using implicit differentiation for the equation sin(x/y) = 1/2 and find dy/dx.

Solution

To solve the problem, we need to find dydx\frac{dy}{dx} using implicit differentiation for the given equation:

sin(xy)=12\sin\left(\frac{x}{y}\right) = \frac{1}{2}

Step 1: Differentiate both sides with respect to xx.

Using implicit differentiation, differentiate sin(xy)\sin\left(\frac{x}{y}\right) with respect to xx, while considering that yy is a function of xx:

cos(xy)(ddx(xy))=0\cos\left(\frac{x}{y}\right) \cdot \left(\frac{d}{dx} \left(\frac{x}{y}\right)\right) = 0

Now, find ddx(xy)\frac{d}{dx}\left(\frac{x}{y}\right) using the quotient rule:

ddx(xy)=y1xdydxy2=yxdydxy2\frac{d}{dx}\left(\frac{x}{y}\right) = \frac{y \cdot 1 - x \cdot \frac{dy}{dx}}{y^2} = \frac{y - x \frac{dy}{dx}}{y^2}

So, substituting this back, we get:

cos(xy)yxdydxy2=0\cos\left(\frac{x}{y}\right) \cdot \frac{y - x \frac{dy}{dx}}{y^2} = 0

Step 2: Solve for dydx\frac{dy}{dx}.

Since cos(xy)0\cos\left(\frac{x}{y}\right) \neq 0, we can equate the numerator to zero:

yxdydx=0y - x \frac{dy}{dx} = 0

This simplifies to:

xdydx=yx \frac{dy}{dx} = y

Thus,

dydx=yx\frac{dy}{dx} = \frac{y}{x}

The solution is:

dydx=yx\frac{dy}{dx} = \frac{y}{x}

Would you like further details or have any other questions?

Related Questions:

  1. What is implicit differentiation, and how does it differ from explicit differentiation?
  2. How can we apply the chain rule in implicit differentiation?
  3. What is the quotient rule, and how is it used in differentiation?
  4. Why do we need to differentiate both sides of the equation with respect to xx?
  5. How can you solve implicit differentiation problems where multiple terms contain yy?

Tip:

When dealing with implicit differentiation, always remember to treat yy as a function of xx, applying the chain rule appropriately.

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Math Problem Analysis

Mathematical Concepts

Implicit Differentiation
Trigonometric Functions
Quotient Rule
Chain Rule

Formulas

sin(x/y) = 1/2
Quotient rule: (d/dx)(u/v) = (v * du/dx - u * dv/dx) / v^2
Chain rule: d/dx[f(g(x))] = f'(g(x)) * g'(x)

Theorems

Differentiation Theorem
Trigonometric Derivatives

Suitable Grade Level

College-level Calculus (First-year)