Question

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

Answer 1

`\displaystyle (dy)/(dx) = (-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 (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Implicit Differentiation

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