Math Problem Statement
найти производную второго порядка
Solution
To solve this, we are given parametric equations:
We need to find the second derivative, .
Step-by-Step Solution:
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Find :
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Find :
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Find : Using the chain rule: Simplify:
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Find the second derivative : Differentiate with respect to and divide by :
First, compute the derivative of : Use the product rule: Simplify:
Combine with :
Simplify:
Let me know if you'd like a further breakdown or have specific questions! Here are some related questions for exploration:
- How is related to the parametric representation of a curve?
- What are the conditions for to be undefined?
- Can this method be generalized to other parametric equations?
- How does the second derivative relate to the concavity of the curve?
- What are the geometrical implications of higher-order derivatives for parametric equations?
Tip: Always simplify trigonometric terms wherever possible to make derivatives easier to interpret.
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Math Problem Analysis
Mathematical Concepts
Parametric Equations
Calculus
Second Derivatives
Trigonometry
Formulas
x = 1 / sin(t), y = tan(t)
dx/dt = -cos(t) / sin^2(t)
dy/dt = sec^2(t)
dy/dx = (dy/dt) / (dx/dt)
d^2y/dx^2 = (d(dy/dx)/dt) / (dx/dt)
Theorems
Chain Rule
Product Rule
Suitable Grade Level
Undergraduate Calculus or Advanced High School (Grades 11-12)
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