Question

Derivative of y = cos(x-1)/(x-1)

Answer 1

`\displaystyle y' = (- \cos (x - 1))/((x - 1)^2) - (\sin (x - 1))/(x - 1)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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 [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)`

Explanation:

Step 1: Define

Identify

`\displaystyle y = (\cos (x - 1))/(x - 1)`

Step 2: Differentiate

1. Derivative Rule [Quotient Rule]:
   `\displaystyle y' = (\Big( \cos (x - 1) \Big)'(x - 1) - \cos (x - 1)(x - 1)')/((x - 1)^2)`
2. Trigonometric Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y' = (- \sin (x - 1)(x - 1)'(x - 1) - \cos (x - 1)(x - 1)')/((x - 1)^2)`
3. Basic Power Rule [Derivative Properties]:
   `\displaystyle y' = (- \sin (x - 1)(x - 1) - \cos (x - 1))/((x - 1)^2)`
4. Simplify:
   `\displaystyle y' = (- \cos (x - 1))/((x - 1)^2) - (\sin (x - 1))/(x - 1)`

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

Unit: Differentiation