Help! Stuck on this question (calculus) If y=(3)/(x^2) work out (dx)/(dy)

Question

Help! Stuck on this question (calculus)

If
`y=(3)/(x^2)`


work out
`(dx)/(dy)`

Answer 1

`\displaystyle (dx)/(dy) = \frac{3\sqrt{(3)/(y)}(y - 1)}{6y}`

General Formulas and Concepts:

Algebra I

• Terms/Coefficients
• Functions
• Function Notation

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Implicit Differentiation

Explanation:

Step 1: Define

Identify

`\displaystyle y = (3)/(x^2)`

Step 2: Differentiate

1. Rewrite Function:
   `\displaystyle yx^2 = 3`
2. [Implicit Differentiation] Basic Power Rule [Product Rule]:
   `\displaystyle y^(1 - 1)x^2 + y(2x^(2 - 1))(dx)/(dy) = 3`
3. Simplify:
   `\displaystyle x^2 + y(2x)(dx)/(dy) = 3`
4. Isolate
   `\displaystyle (dx)/(dy)` term:
   `\displaystyle y(2x)(dx)/(dy) = 3 - x^2`
5. Isolate
   `\displaystyle (dx)/(dy)`:
   `\displaystyle (dx)/(dy) = (3 - x^2)/(2xy)`
6. Substitute in x²:
   `\displaystyle (dx)/(dy) = (3 - (3)/(y))/(2xy)`
7. Simplify:
   `\displaystyle (dx)/(dy) = (3(y - 1))/(2xy^2)`
8. Substitute in x:
   `\displaystyle (dx)/(dy) = \frac{3(y - 1)}{2y^2(\sqrt{(3)/(y)})}`
9. Simplify:
   `\displaystyle (dx)/(dy) = \frac{3\sqrt{(3)/(y)}(y - 1)}{6y}`

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

Unit: Derivatives

Book: College Calculus 10e