Mathematics, 22.03.2021 01:00 majestyic1207

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

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Answer from: u8p4

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

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition 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]: $\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)$

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:

Step 1: Define

$\displaystyle y = ln(x^2 + 6)^{\frac{3}{2}}$

Step 2: Differentiate

[Derivative] Chain Rule: $\displaystyle y' = \frac{d}{dx}[ln(x^2 + 6)^{\frac{3}{2}}] \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]$ [Derivative] Chain Rule [Basic Power Rule]: $\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{3}{2} - 1} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]$ [Derivative] Simplify: $\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{d}{dx}[ln(x^2 + 6)] \cdot \frac{d}{dx}[x^2 + 6]$ [Derivative] ln Derivative: $\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot \frac{d}{dx}[x^2 + 6]$ [Derivative] Basic Power Rule: $\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot (2 \cdot x^{2 - 1} + 0)$ [Derivative] Simplify: $\displaystyle y' = \frac{3}{2}ln(x^2 + 6)^{\frac{1}{2}} \cdot \frac{1}{x^2 + 6} \cdot (2x)$ [Derivative] Multiply: $\displaystyle y' = \frac{3ln(x^2 + 6)^{\frac{1}{2}}}{2} \cdot \frac{1}{x^2 + 6} \cdot (2x)$ [Derivative] Multiply: $\displaystyle y' = \frac{3ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)} \cdot (2x)$ [Derivative] Multiply: $\displaystyle y' = \frac{3(2x)ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}$ [Derivative] Multiply: $\displaystyle y' = \frac{6xln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}$ [Derivative] Factor: $\displaystyle y' = \frac{2(3x)ln(x^2 + 6)^{\frac{1}{2}}}{2(x^2 + 6)}$ [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

Answer from: Quest

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