Question

The problem is attached, thanks.

Answer 1

`\displaystyle (dy)/(dx) \bigg| \limit_((1, 4)) = 2`

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]:
  `\displaystyle \sqrt[n]{x} = x^{(1)/(n)}`
• Exponential Rule [Rewrite]:
  `\displaystyle b^(-m) = (1)/(b^m)`

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Implicit Differentiation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Explanation:

Step 1: Define

`\displaystyle √(x) - √(y) = -1`

Point (1, 4)

Step 2: Differentiate

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

Step 3: Evaluate

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

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

Unit: Implicit Differentiation

Book: College Calculus 10e