Question

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

Answer 1

Answer:

$\displaystyle y' = \frac{-3x^2}{10y}$

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:                                                           $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

Derivative Property [Addition/Subtraction]:                                                         $\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)]$  

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Chain Rule]:                                                                                 $\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)$

Implicit Differentiation

Step-by-step explanation:

Step 1: Define

Identify

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

Step 2: Differentiate

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

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

Unit: Differentiation