Answer:
D. undefined
General Formulas and Concepts:
Calculus
Derivatives
Derivative Notation
Derivative of a constant is 0
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Derivative Rule [Chain Rule]: [tex]\displaystyle \frac{d}{dx}[f(g(x))] =f'(g(x)) \cdot g'(x)[/tex]
Trig Derivative: [tex]\displaystyle \frac{d}{dx}[sinu] = u'cosu[/tex]
Derivatives of Parametrics: [tex]\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}[/tex]
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle \frac{dx}{dt} = 5[/tex]
[tex]\displaystyle \frac{dy}{dt} = sin(t^2)[/tex]
Step 2: Differentiate
- [x Derivative] Basic Power Rule: [tex]\displaystyle \frac{d^2x}{dt^2} = 0[/tex]
- [y Derivative] Trig Derivative [Chain Rule]: [tex]\displaystyle \frac{d^2y}{dt^2} = cos(t^2) \cdot \frac{d}{dt}[t^2][/tex]
- [y Derivative] Basic Power Rule: [tex]\displaystyle \frac{d^2y}{dt^2} = cos(t^2) \cdot 2t^{2 - 1}[/tex]
- [y Derivative] Simplify: [tex]\displaystyle \frac{d^2y}{dt^2} = 2tcos(t^2)[/tex]
- [Derivative] Rewrite: [tex]\displaystyle \frac{d^2y}{dx^2} = \frac{2tcos(t^2)}{0}[/tex]
Anything divided by 0 is undefined.
Topic: AP Calculus BC (Calculus I/II)
Unit: Differentiation with Parametrics
Book: College Calculus 10e
