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MrRa [10]
3 years ago
12

Find y' if y= In (x2 +6)^3/2 y'=

Mathematics
1 answer:
Schach [20]3 years ago
3 0

Answer:

\displaystyle y' = \frac{3xln(x^2 + 6)^{\frac{1}{2}}}{x^2 + 6}

General Formulas and Concepts:

<u>Pre-Algebra</u>

Order of Operations: BPEMDAS

  1. Brackets
  2. Parenthesis
  3. Exponents
  4. Multiplication
  5. Division
  6. Addition
  7. Subtraction
  • Left to Right<u> </u>

<u>Algebra I</u>

  • Factoring

<u>Calculus</u>

Derivatives

Derivative Notation

Derivative of a constant is 0

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

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

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

ln Derivative: \displaystyle \frac{d}{dx} [lnu] = \frac{u'}{u}

Step-by-step explanation:

<u>Step 1: Define</u>

<u />\displaystyle y = ln(x^2 + 6)^{\frac{3}{2}}<u />

<u />

<u>Step 2: Differentiate</u>

  1. [Derivative] Chain Rule:                                                                                 \displaystyle y' = \frac{d}{dx}[ln(x^2 + 6)^{\frac{3}{2}}] \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]
  2. [Derivative] Chain Rule [Basic Power Rule]:                                                 \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{3}{2} - 1} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]
  3. [Derivative] Simplify:                                                                                      \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]
  4. [Derivative] ln Derivative:                                                                               \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot \frac{d}{dx}[x^2 + 6]
  5. [Derivative] Basic Power Rule:                                                                      \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot (2 \cdot x^{2 - 1} + 0)
  6. [Derivative] Simplify:                                                                                       \displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot (2x)
  7. [Derivative] Multiply:                                                                                       \displaystyle y' = \frac{3ln(x^2 + 6)^{\frac{1}{2}}}{2} \cdot \frac{1}{x^2 + 6} \cdot (2x)
  8. [Derivative] Multiply:                                                                                       \displaystyle y' = \frac{3ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)} \cdot (2x)
  9. [Derivative] Multiply:                                                                                       \displaystyle y' = \frac{3(2x)ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}
  10. [Derivative] Multiply:                                                                                       \displaystyle y' = \frac{6xln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}
  11. [Derivative] Factor:                                                                                         \displaystyle y' = \frac{2(3x)ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}
  12. [Derivative] Simplify:                                                                                       \displaystyle y' = \frac{3xln(x^2 + 6)^{\frac{1}{2}}}{x^2 + 6}

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

Unit: Derivatives

Book: College Calculus 10e

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