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iris [78.8K]
3 years ago
8

Differentiate the given function:​

Mathematics
1 answer:
Natasha2012 [34]3 years ago
6 0

Answer:

\displaystyle h'(x) = \frac{1 + x - arcsin(x)\sqrt{1 - x^2}}{\sqrt{1 - x^2}(1 + x)^2}

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>Algebra I</u>

  • Terms/Coefficients
  • Functions
  • Function Notation

<u>Algebra II</u>

  • Simplifying

<u>Calculus</u>

Derivatives

Derivative Notation

Basic Power Rule:

  • f(x) = cxⁿ
  • f’(x) = c·nxⁿ⁻¹

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

Derivative Rule [Quotient Rule]:                                                                           \displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}

Special Trig Derivatives:

  • Arcsine:                                                                                                         \displaystyle \frac{d}{dx}[arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}

Step-by-step explanation:

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle h(x) = \frac{arcsin(x)}{1 + x}

<u>Step 2: Differentiate</u>

  1. Quotient Rule:                                                                                               \displaystyle h'(x) = \frac{(1 + x)\frac{d}{dx}[arcsin(x)] - \frac{d}{dx}[1 + x][arcsin(x)]}{(1 + x)^2}
  2. Special Trig Derivative [Arcsine]:                                                                 \displaystyle h'(x) = \frac{(1 + x)(\frac{1}{\sqrt{1 - x^2}}) - \frac{d}{dx}[1 + x][arcsin(x)]}{(1 + x)^2}
  3. Derivative Property [Addition/Subtraction]:                                                 \displaystyle h'(x) = \frac{(1 + x)(\frac{1}{\sqrt{1 - x^2}}) - (\frac{d}{dx}[1] + \frac{d}{dx}[x])[arcsin(x)]}{(1 + x)^2}
  4. Basic Power Rule:                                                                                         \displaystyle h'(x) = \frac{(1 + x)(\frac{1}{\sqrt{1 - x^2}}) - 1[arcsin(x)]}{(1 + x)^2}
  5. Multiply:                                                                                                         \displaystyle h'(x) = \frac{\frac{1 + x}{\sqrt{1 - x^2}} - arcsin(x)}{(1 + x)^2}
  6. Rewrite [Multiply]:                                                                                         \displaystyle h'(x) = \frac{\frac{1 + x}{\sqrt{1 - x^2}} - arcsin(x)}{(1 + x)^2} \cdot \frac{\sqrt{1 - x^2}}{\sqrt{1 - x^2}}
  7. Simplify [Multiply]:                                                                                         \displaystyle h'(x) = \frac{1 + x - arcsin(x)\sqrt{1 - x^2}}{\sqrt{1 - x^2}(1 + x)^2}

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

Unit: Derivatives

Book: College Calculus 10e

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