Math Problem Statement
Kerjakan
Solution
Let's go through the problems one by one.
1. Derivatives of
To calculate the first derivative :
2. Implicit Differentiation of
To find , differentiate both sides with respect to :
3. Velocity from Position Function
To find the velocity, we need to differentiate the position function : At : Thus, the velocity at is .
4. Chain Rule:
Using the chain rule, let , so: First, calculate : Now substitute back:
5. Second Derivative of
The first derivative: Now, the second derivative: The second derivative describes the acceleration or change in the rate of the angular velocity in the context of rotational motion.
Do you want more details on any specific step or explanation?
Here are 5 related questions for practice:
- What is the second derivative of ?
- If , what is in terms of and ?
- What is the acceleration for the position function ?
- Find the derivative of using the chain rule.
- What is the third derivative of ?
Tip: When working with implicit differentiation, remember to apply the chain rule when differentiating terms involving , as is a function of .
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Math Problem Analysis
Mathematical Concepts
Derivatives
Implicit Differentiation
Velocity from Position Function
Chain Rule
Second Derivative
Formulas
f'(x) = 4x^3 - 9x^2 + 4x - 5
Implicit Differentiation: dy/dx = -x/y
Velocity: v(t) = ds/dt = 9t^2 - 12t + 9
Chain Rule: f'(x) = 7(3x^2 - 4x + 5)^6 * (6x - 4)
Second Derivative: f''(x) = -9 sin(3x)
Theorems
Power Rule for Derivatives
Implicit Differentiation
Chain Rule
Suitable Grade Level
University Level (Calculus I or II)
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