Question

Differentiate the given function:​

Answer 1

`\displaystyle h'(x) = (1 + x - arcsin(x)√(1 - x^2))/(√(1 - x^2)(1 + x)^2)`

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

• Terms/Coefficients
• Functions
• Function Notation

Algebra II

• Simplifying

Calculus

Derivatives

Derivative Notation

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

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

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

Special Trig Derivatives:

• Arcsine:
  `\displaystyle (d)/(dx)[arcsin(x)] = (1)/(√(1 - x^2))`

Explanation:

Step 1: Define

Identify

`\displaystyle h(x) = (arcsin(x))/(1 + x)`

Step 2: Differentiate

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

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

Unit: Derivatives

Book: College Calculus 10e