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BlackZzzverrR [31]
3 years ago
11

Find y' for the following.​

Mathematics
1 answer:
Varvara68 [4.7K]3 years ago
4 0

Answer:

\displaystyle y' = \frac{5x - 2xy^2}{2y(x^2 - 3y)}

General Formulas and Concepts:

<u>Calculus</u>

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 [Product Rule]:                                                                             \displaystyle \frac{d}{dx} [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)

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

Implicit Differentiation

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

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

<u>Step 2: Differentiate</u>

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

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

Unit: Differentiation

Book: College Calculus 10e

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