Question

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

Answer 1

Answer:

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

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]:                                                                                       $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

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

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

Unit: Implicit Differentiation

Book: College Calculus 10e