Question

What is the derivative of cos(pi*t/4) ?

Answer 1

`\displaystyle (dy)/(dx) = (-\pi)/(4) \sin \bigg( (\pi t)/(4) \bigg)`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

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

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = \cos \bigg( (\pi t)/(4) \bigg)`

Step 2: Differentiate

1. Trigonometric Differentiation [Derivative Rule - Chain Rule]:
   `\displaystyle y' = -\sin \bigg( (\pi t)/(4) \bigg) \cdot (d)/(dt) \bigg[ (\pi t)/(4) \bigg]`
2. Rewrite [Derivative Property - Multiplied Constant]:
   `\displaystyle y' = -\sin \bigg( (\pi t)/(4) \bigg) \cdot (\pi)/(4) (d)/(dt)[t]`
3. Basic Power Rule:
   `\displaystyle y' = (-\pi)/(4)\sin \bigg( (\pi t)/(4) \bigg)`

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

Unit: Differentiation