Respuesta :

Space

Answer:

[tex]\displaystyle \frac{dy}{dx} \bigg| \limit_{(1, 4)} = 2[/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

  • Multiplication Property of Equality
  • Division Property of Equality
  • Addition Property of Equality
  • Subtraction Property of Equality

Algebra I

  • Coordinates (x, y)
  • Exponential Rule [Root Rewrite]:                                                                 [tex]\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}[/tex]
  • Exponential Rule [Rewrite]:                                                                           [tex]\displaystyle b^{-m} = \frac{1}{b^m}[/tex]

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Implicit Differentiation

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

Step-by-step explanation:

Step 1: Define

[tex]\displaystyle \sqrt{x} - \sqrt{y} = -1[/tex]

Point (1, 4)

Step 2: Differentiate

  1. [Function] Rewrite [Exponential Rule - Root Rewrite]:                               [tex]\displaystyle x^{\frac{1}{2}} - y^{\frac{1}{2}} = -1[/tex]
  2. [Implicit Differentiation] Basic Power Rule:                                                 [tex]\displaystyle \frac{1}{2}x^{\frac{1}{2} - 1} - \frac{1}{2}y^{\frac{1}{2} - 1}\frac{dy}{dx} = 0[/tex]
  3. [Implicit Differentiation] Simplify Exponents:                                               [tex]\displaystyle \frac{1}{2}x^{\frac{-1}{2}} - \frac{1}{2}y^{\frac{-1}{2}}\frac{dy}{dx} = 0[/tex]
  4. [Implicit Differentiation] Rewrite [Exponential Rule - Rewrite]:                   [tex]\displaystyle \frac{1}{2x^{\frac{1}{2}}} - \frac{1}{2y^{\frac{1}{2}}}\frac{dy}{dx} = 0[/tex]
  5. [Implicit Differentiation] Isolate y terms:                                                       [tex]\displaystyle -\frac{1}{2y^{\frac{1}{2}}}\frac{dy}{dx} = -\frac{1}{2x^{\frac{1}{2}}}[/tex]
  6. [Implicit Differentiation] Isolate [tex]\displaystyle \frac{dy}{dx}[/tex]:                                                               [tex]\displaystyle \frac{dy}{dx} = \frac{2y^{\frac{1}{2}}}{2x^{\frac{1}{2}}}[/tex]
  7. [Implicit Differentiation] Simplify:                                                                 [tex]\displaystyle \frac{dy}{dx} = \frac{y^{\frac{1}{2}}}{x^{\frac{1}{2}}}[/tex]

Step 3: Evaluate

  1. Substitute in point [Derivative]:                                                                     [tex]\displaystyle \frac{dy}{dx} = \frac{(4)^{\frac{1}{2}}}{(1)^{\frac{1}{2}}}[/tex]
  2. Exponents:                                                                                                     [tex]\displaystyle \frac{dy}{dx} = \frac{2}{1}[/tex]
  3. Division:                                                                                                         [tex]\displaystyle \frac{dy}{dx} = 2[/tex]

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

Unit: Implicit Differentiation

Book: College Calculus 10e

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