Question

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

Answer 1

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

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Factoring

Calculus

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 (d)/(dx) [cf(x)] = c \cdot f'(x)`

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

ln Derivative:
`\displaystyle (d)/(dx) [lnu] = (u')/(u)`

Explanation:

Step 1: Define

`\displaystyle y = ln(x^2 + 6)^{(3)/(2)}`

Step 2: Differentiate

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

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

Unit: Derivatives

Book: College Calculus 10e