Question

Derivative of x^3 + 5y^2=2

Answer 1

`\displaystyle y' = (-3x^2)/(10y)`

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

Implicit Differentiation

Explanation:

Step 1: Define

Identify

`\displaystyle x^3 + 5y^2 = 2`

Step 2: Differentiate

1. Implicit Differentiation [Derivative Property - Addition/Subtraction]:
   `\displaystyle (d)/(dx)[x^3] + (d)/(dx)[5y^2] = (d)/(dx)[2]`
2. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle (d)/(dx)[x^3] + 5 (d)/(dx)[y^2] = (d)/(dx)[2]`
3. Basic Power Rule [Derivative Rule - Chain Rule]:
   `\displaystyle 3x^2 + 10yy' = 0`
4. Isolate y' term:
   `\displaystyle 10yy' = -3x^2`
5. Isolate y':
   `\displaystyle y' = (-3x^2)/(10y)`

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

Unit: Differentiation