Answer:
\(\displaystyle \frac{dy}{dx} = \frac{-\pi}{4} \sin \bigg( \frac{\pi t}{4} \bigg)\)
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Multiplied Constant]: \(\displaystyle \frac{d}{dx} [cf(x)] = c \cdot f'(x)\)
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Chain Rule]: \(\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)\)
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle y = \cos \bigg( \frac{\pi t}{4} \bigg)\)
Step 2: Differentiate
- Trigonometric Differentiation [Derivative Rule - Chain Rule]: \(\displaystyle y' = -\sin \bigg( \frac{\pi t}{4} \bigg) \cdot \frac{d}{dt} \bigg[ \frac{\pi t}{4} \bigg]\)
- Rewrite [Derivative Property - Multiplied Constant]: \(\displaystyle y' = -\sin \bigg( \frac{\pi t}{4} \bigg) \cdot \frac{\pi}{4} \frac{d}{dt}[t]\)
- Basic Power Rule: \(\displaystyle y' = \frac{-\pi}{4}\sin \bigg( \frac{\pi t}{4} \bigg)\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Differentiation
