Question

Find y' for the following.​

Answer 1

`\displaystyle y' = (5x - 2xy^2)/(2y(x^2 - 3y))`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

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)`

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

Implicit Differentiation

Explanation:

Step 1: Define

Identify

`\displaystyle 5x^2 - 2x^2y^2 + 4y^3 - 7 = 0`

Step 2: Differentiate

1. Implicit Differentiation:
   `\displaystyle (dy)/(dx)[5x^2 - 2x^2y^2 + 4y^3 - 7] = (dy)/(dx)[0]`
2. Rewrite [Derivative Property - Addition/Subtraction]:
   `\displaystyle (dy)/(dx)[5x^2] - (dy)/(dx)[2x^2y^2] + (dy)/(dx)[4y^3] - (dy)/(dx)[7] = (dy)/(dx)[0]`
3. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle 5(dy)/(dx)[x^2] - 2(dy)/(dx)[x^2y^2] + 4(dy)/(dx)[y^3] - (dy)/(dx)[7] = (dy)/(dx)[0]`
4. Basic Power Rule [Product Rule, Chain Rule]:
   `\displaystyle 10x - 2 \Big( (d)/(dx)[x^2]y^2 + x^2(d)/(dx)[y^2] \Big) + 12y^2y' - 0 = 0`
5. Basic Power Rule [Chain Rule]:
   `\displaystyle 10x - 2 \Big( 2xy^2 + x^22yy' \Big) + 12y^2y' - 0 = 0`
6. Simplify:
   `\displaystyle 10x - 4xy^2 - 4x^2yy' + 12y^2y' = 0`
7. Isolate y' terms:
   `\displaystyle -4x^2yy' + 12y^2y' = 4xy^2 - 10x`
8. Factor:
   `\displaystyle y'(-4x^2y + 12y^2) = 4xy^2 - 10x`
9. Isolate y':
   `\displaystyle y' = (4xy^2 - 10x)/(-4x^2y + 12y^2)`
10. Simplify:
    `\displaystyle y' = (5x - 2xy^2)/(2y(x^2 - 3y))`

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

Unit: Differentiation

Book: College Calculus 10e