Question

Differentiate the following Functions
5x^2-2xy + 4y^3= 5

Answer 1

`\displaystyle y' = (y - 5x)/(x + 6y^2)`

General Formulas and Concepts:

Algebra I

• Terms/Coefficients
• Factoring

Calculus

Differentiation

• Derivatives
• Derivative Notation
• Implicit Differentiation

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)`

Explanation:

Step 1: Define

Identify

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

Step 2: Differentiate

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

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

Unit: Differentiation

Book: College Calculus 10e