Answer:
[tex]\displaystyle h'(x) = \frac{1 + x - arcsin(x)\sqrt{1 - x^2}}{\sqrt{1 - x^2}(1 + x)^2}[/tex]
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
- Brackets
- Parenthesis
- Exponents
- Multiplication
- Division
- Addition
- Subtraction
Algebra I
- Terms/Coefficients
- Functions
- Function Notation
Algebra II
Calculus
Derivatives
Derivative Notation
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Derivative Rule [Quotient Rule]: [tex]\displaystyle \frac{d}{dx} [\frac{f(x)}{g(x)} ]=\frac{g(x)f'(x)-g'(x)f(x)}{g^2(x)}[/tex]
Special Trig Derivatives:
- Arcsine: [tex]\displaystyle \frac{d}{dx}[arcsin(x)] = \frac{1}{\sqrt{1 - x^2}}[/tex]
Step-by-step explanation:
Step 1: Define
Identify
[tex]\displaystyle h(x) = \frac{arcsin(x)}{1 + x}[/tex]
Step 2: Differentiate
- Quotient Rule: [tex]\displaystyle h'(x) = \frac{(1 + x)\frac{d}{dx}[arcsin(x)] - \frac{d}{dx}[1 + x][arcsin(x)]}{(1 + x)^2}[/tex]
- Special Trig Derivative [Arcsine]: [tex]\displaystyle h'(x) = \frac{(1 + x)(\frac{1}{\sqrt{1 - x^2}}) - \frac{d}{dx}[1 + x][arcsin(x)]}{(1 + x)^2}[/tex]
- Derivative Property [Addition/Subtraction]: [tex]\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}[/tex]
- Basic Power Rule: [tex]\displaystyle h'(x) = \frac{(1 + x)(\frac{1}{\sqrt{1 - x^2}}) - 1[arcsin(x)]}{(1 + x)^2}[/tex]
- Multiply: [tex]\displaystyle h'(x) = \frac{\frac{1 + x}{\sqrt{1 - x^2}} - arcsin(x)}{(1 + x)^2}[/tex]
- Rewrite [Multiply]: [tex]\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}}[/tex]
- Simplify [Multiply]: [tex]\displaystyle h'(x) = \frac{1 + x - arcsin(x)\sqrt{1 - x^2}}{\sqrt{1 - x^2}(1 + x)^2}[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Derivatives
Book: College Calculus 10e
