Question

If the gradient of the tangent to


`y = √(x)`
is

`(1)/(6)`
at point A, find the coordinates of A.​

Answer 1

Point A(9, 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

• Coordinates (x, y)
• Functions
• Function Notation
• Terms/Coefficients
• Anything to the 0th power is 1
• 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

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Explanation:

Step 1: Define

Identify

`\displaystyle y = √(x)`

`\displaystyle y' = (1)/(6)`

Step 2: Differentiate

1. [Function] Rewrite [Exponential Rule - Root Rewrite]:
   `\displaystyle y = x^{(1)/(2)}`
2. Basic Power Rule:
   `\displaystyle y' = (1)/(2)x^{(1)/(2) - 1}`
3. Simplify:
   `\displaystyle y' = (1)/(2)x^{-(1)/(2)}`
4. [Derivative] Rewrite [Exponential Rule - Rewrite]:
   `\displaystyle y' = \frac{1}{2x^{(1)/(2)}}`
5. [Derivative] Rewrite [Exponential Rule - Root Rewrite]:
   `\displaystyle y' = (1)/(2√(x))`

Step 3: Solve

Find coordinates of A.

x-coordinate

1. Substitute in y' [Derivative]:
   `\displaystyle (1)/(6) = (1)/(2√(x))`
2. [Multiplication Property of Equality] Multiply 2 on both sides:
   `\displaystyle (1)/(3) = (1)/(√(x))`
3. [Multiplication Property of Equality] Cross-multiply:
   `\displaystyle √(x) = 3`
4. [Equality Property] Square both sides:
   `\displaystyle x = 9`

y-coordinate

1. Substitute in x [Function]:
   `\displaystyle y = √(9)`
2. [√Radical] Evaluate:
   `\displaystyle y = 3`

∴ Coordinates of A is (9, 3).

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

Unit: Derivatives

Book: College Calculus 10e