Question

The problem is attached, thanks.

Answer 1

Answer:

$\displaystyle \frac{dy}{dx} \bigg| \limit_{(1, 4)} = 2$

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]:                                                                 $\displaystyle \sqrt[n]{x} = x^{\frac{1}{n}}$
• Exponential Rule [Rewrite]:                                                                           $\displaystyle b^{-m} = \frac{1}{b^m}$

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

Implicit Differentiation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Step-by-step explanation:

Step 1: Define

$\displaystyle \sqrt{x} - \sqrt{y} = -1$

Point (1, 4)

Step 2: Differentiate

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

Step 3: Evaluate

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

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

Unit: Implicit Differentiation

Book: College Calculus 10e