Question

Help with num 3 please. thanks​

Answer 1

a)
`\displaystyle (dy)/(dx) \bigg| \limits_(x = 0) = -1`

b)
`\displaystyle (dy)/(dx) \bigg| \limits_{x = (\pi)/(2)} = -1`

General Formulas and Concepts:

Pre-Calculus

• Unit Circle

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
`\displaystyle (d)/(dx) [cf(x)] = c \cdot f'(x)`

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

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Derivative Rule [Product Rule]:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

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

Derivative Rule [Chain Rule]:
`\displaystyle (d)/(dx)[f(g(x))] =f'(g(x)) \cdot g'(x)`

Trigonometric Differentiation

Logarithmic Differentiation

Explanation:

a)

Step 1: Define

Identify

`\displaystyle y = ln \bigg( (1 - x)/(√(1 + x^2)) \bigg)`

Step 2: Differentiate

1. Logarithmic Differentiation [Chain Rule]:
   `\displaystyle (dy)/(dx) = (1)/((1 - x)/(√(1 + x^2))) \cdot (d)/(dx)[(1 - x)/(√(1 + x^2))]`
2. Simplify:
   `\displaystyle (dy)/(dx) = (-√(x^2 + 1))/(x - 1) \cdot (d)/(dx)[(1 - x)/(√(1 + x^2))]`
3. Quotient Rule:
   `\displaystyle (dy)/(dx) = (-√(x^2 + 1))/(x - 1) \cdot ((1 - x)'√(1 + x^2) - (1 - x)(√(1 + x^2))')/((√(1 + x^2))^2)`
4. Basic Power Rule [Chain Rule]:
   `\displaystyle (dy)/(dx) = (-√(x^2 + 1))/(x - 1) \cdot (-√(1 + x^2) - (1 - x)((x)/(√(x^2 + 1))))/((√(1 + x^2))^2)`
5. Simplify:
   `\displaystyle (dy)/(dx) = (-√(x^2 + 1))/(x - 1) \cdot \bigg( \frac{x(x - 1)}{(x^2 + 1)^\bigg{(3)/(2)}} - (1)/(√(x^2 + 1)) \bigg)`
6. Simplify:
   `\displaystyle (dy)/(dx) = (x + 1)/((x - 1)(x^2 + 1))`

Step 3: Find

1. Substitute in x = 0 [Derivative]:
   `\displaystyle (dy)/(dx) \bigg| \limit_(x = 0) = (0 + 1)/((0 - 1)(0^2 + 1))`
2. Evaluate:
   `\displaystyle (dy)/(dx) \bigg| \limits_(x = 0) = -1`

b)

Step 1: Define

Identify

`\displaystyle y = ln \bigg( (1 + sinx)/(1 - cosx) \bigg)`

Step 2: Differentiate

1. Logarithmic Differentiation [Chain Rule]:
   `\displaystyle (dy)/(dx) = (1)/((1 + sinx)/(1 - cosx)) \cdot (d)/(dx)[(1 + sinx)/(1 - cosx)]`
2. Simplify:
   `\displaystyle (dy)/(dx) = (-[cos(x) - 1])/(sin(x) + 1) \cdot (d)/(dx)[(1 + sinx)/(1 - cosx)]`
3. Quotient Rule:
   `\displaystyle (dy)/(dx) = (-[cos(x) - 1])/(sin(x) + 1) \cdot ((1 + sinx)'(1 - cosx) - (1 + sinx)(1 - cosx)')/((1 - cosx)^2)`
4. Trigonometric Differentiation:
   `\displaystyle (dy)/(dx) = (-[cos(x) - 1])/(sin(x) + 1) \cdot (cos(x)(1 - cosx) - sin(x)(1 + sinx))/((1 - cosx)^2)`
5. Simplify:
   `\displaystyle (dy)/(dx) = (-[cos(x) - sin(x) - 1])/([sin(x) + 1][cos(x) - 1])`

Step 3: Find

1. Substitute in x = π/2 [Derivative]:
   `\displaystyle (dy)/(dx) \bigg| \limit_{x = (\pi)/(2)} = (-[cos((\pi)/(2)) - sin((\pi)/(2)) - 1])/([sin((\pi)/(2)) + 1][cos((\pi)/(2)) - 1])`
2. Evaluate [Unit Circle]:
   `\displaystyle (dy)/(dx) \bigg| \limit_{x = (\pi)/(2)} = -1`

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

Unit: Differentiation

Book: College Calculus 10e