Respuesta :

Space

Answer:

[tex]\frac{dy}{dx} = \frac{sin(y) + ycos(x)}{sin(x) - xcos(y)}[/tex]

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right

Equality Properties

Algebra I

  • Factoring

Calculus

  • Derivatives
  • Derivative Notation
  • Implicit Differentiation
  • Trig Derivative: [tex]\frac{d}{dx} [sin(u)] = cos(u) \cdot u'[/tex]
  • Product Rule: [tex]\frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)[/tex]

Step-by-step explanation:

Step 1: Define

[tex]ysin(x) = xsin(y)[/tex]

Step 2: Differentiate

Implicit Differentiation

  1. Differentiate [Product Rule/Trig}:                                                                  [tex]y'sin(x) + ycos(x) = sin(y) + xcos(y)y'[/tex]
  2. Subtract ycos(x) on both sides:                                                                      [tex]y'sin(x) = sin(y) + xy'cos(y) - ycos(x)[/tex]
  3. Subtract xy'cos(y) on both sides:                                                                     [tex]y'sin(x) - xy'cos(y) = sin(y) - ycos(x)[/tex]
  4. Factor out y':                                                                                                   [tex]y'[sin(x) - xcos(y)] = sin(y) - ycos(x)[/tex]
  5. Isolate y':                                                                                                            [tex]y' = \frac{sin(y) - ycos(x)}{sin(x) - xcos(y)}[/tex]