Question

What is the slope of the line tangent to the curve square root (x) +square root (y) = 2 at the point ( 9/4, 1/4 )? (photo attached of answer choices)

Answer 1

B.
`\displaystyle -(1)/(3)`

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

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

Calculus

Derivatives

Derivative Notation

Derivative of a constant is 0

The definition of a derivative is the slope of the tangent line

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Implicit Differentiation

Explanation:

Step 1: Define

`\displaystyle √(x) + √(y) = 2`

`\displaystyle ((9)/(4), (1)/(4))`

Step 2: Differentiate

Implicit Differentiation

1. [Function] Rewrite [Exponential Rule - Root Rewrite]:
   `\displaystyle x^{(1)/(2)} + y^{(1)/(2)} = 2`
2. [Function] Basic Power Rule:
   `\displaystyle (1)/(2)x^{(1)/(2) - 1} + (1)/(2)y^{(1)/(2) - 1}(dy)/(dx) = 0`
3. [Derivative] Simplify:
   `\displaystyle (1)/(2)x^{(-1)/(2)} + (1)/(2)y^{(-1)/(2)}(dy)/(dx) = 0`
4. [Derivative] Rewrite [Exponential Rule - Rewrite]:
   `\displaystyle \frac{1}{2x^{(1)/(2)}} + \frac{1}{2y^{(1)/(2)}}(dy)/(dx) = 0`
5. [Derivative] Isolate
   `\displaystyle (dy)/(dx)` term [Subtraction Property of Equality]:
   `\displaystyle \frac{1}{2y^{(1)/(2)}}(dy)/(dx) = -\frac{1}{2x^{(1)/(2)}}`
6. [Derivative] Isolate
   `\displaystyle (dy)/(dx)` [Multiplication Property of Equality]:
   `\displaystyle (dy)/(dx) = -\frac{2y^{(1)/(2)}}{2x^{(1)/(2)}}`
7. [Derivative] Simplify:
   `\displaystyle (dy)/(dx) = -\frac{y^{(1)/(2)}}{x^{(1)/(2)}}`

Step 3: Evaluate

Find slope of tangent line

1. Substitute in point [Derivative]:
   `\displaystyle (dy)/(dx) \bigg| \limit_{((9)/(4), (1)/(4))} = -\frac{((1)/(4))^{(1)/(2)}}{((9)/(4))^{(1)/(2)}}`
2. [Slope] Exponents:
   `\displaystyle (dy)/(dx) \bigg| \limit_{((9)/(4), (1)/(4))} = -((1)/(2))/((3)/(2))`
3. [Slope] Simplify:
   `\displaystyle (dy)/(dx) \bigg| \limit_{((9)/(4), (1)/(4))} = -(1)/(3)`

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

Unit: Differentiation - Implicit Differentiation

Book: College Calculus 10e