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Differentiate the given function:​

Differentiate the given function:​-example-1
User Mike Burba
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Answer:


\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

User Chris Nicola
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