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Find a formula for dy/dx if sin x + cos y + sec(xy) = 251

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Answer:

[tex]\displaystyle \frac{dy}{dx} = \frac{-cos(x) - ysec(xy)tan(xy)}{-sin(y) + xsec(xy)tan(xy)}[/tex]

General Formulas and Concepts:

Pre-Algebra

Distributive Property

Algebra I

  • Factoring

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Trig Differentiation

Derivative Rule [Chain Rule]:                                                                                       [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]

Implicit Differentiation

Step-by-step explanation:

Step 1: Define

Identify

sin(x) + cos(y) + sec(xy) = 251

Step 2: Differentiate

  1. [Implicit Differentiation] Trig Differentiation [Chain Rule]:                             [tex]\displaystyle cos(x) - sin(y)\frac{dy}{dx} + sec(xy)tan(xy) \cdot (y + x\frac{dy}{dx}) = 0[/tex]                      
  2. [Subtraction Property of Equality] Isolate  [tex]\displaystyle \frac{dy}{dx}[/tex]  terms:                                     [tex]\displaystyle -sin(y)\frac{dy}{dx} + sec(xy)tan(xy) \cdot (y + x\frac{dy}{dx}) = -cos(x)[/tex]
  3. [Distributive Property] Distribute sec(xy)tan(xy):                                            [tex]\displaystyle -sin(y)\frac{dy}{dx} + ysec(xy)tan(xy) + xsec(xy)tan(xy)\frac{dy}{dx} = -cos(x)[/tex]
  4. [Subtraction Property of Equality] Isolate  [tex]\displaystyle \frac{dy}{dx}[/tex]  terms:                                     [tex]\displaystyle -sin(y)\frac{dy}{dx} + xsec(xy)tan(xy)\frac{dy}{dx} = -cos(x) - ysec(xy)tan(xy)[/tex]
  5. Factor out  [tex]\displaystyle \frac{dy}{dx}[/tex]:                                                                                                   [tex]\displaystyle \frac{dy}{dx}[-sin(y) + xsec(xy)tan(xy)] = -cos(x) - ysec(xy)tan(xy)[/tex]
  6. [Division Property of Equality] Isolate  [tex]\displaystyle \frac{dy}{dx}[/tex]:                                                      [tex]\displaystyle \frac{dy}{dx} = \frac{-cos(x) - ysec(xy)tan(xy)}{-sin(y) + xsec(xy)tan(xy)}[/tex]

Topic: AP Calculus AB/BC (Calculus I/II)

Unit: Implicit Differentiation

Book: College Calculus 10e

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