Math Problem Statement
Tính đạo hàm y'' = y''(x) của hàm số y = y(x) cho bởi phương trình tham số { x = ln(t), y = t^3 }.
Solution
The problem asks to calculate the second derivative for the function , where the relationship between and is given by parametric equations:
x = \ln t \\ y = t^3 \end{cases}$$ ### Solution Steps: 1. **Find $$\frac{dy}{dt}$$ and $$\frac{dx}{dt}$$**: - From $$y = t^3$$, we differentiate with respect to $$t$$: $$\frac{dy}{dt} = 3t^2$$ - From $$x = \ln t$$, we differentiate with respect to $$t$$: $$\frac{dx}{dt} = \frac{1}{t}$$ 2. **Find the first derivative $$\frac{dy}{dx}$$**: Using the chain rule: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{3t^2}{\frac{1}{t}} = 3t^3$$ 3. **Find the second derivative $$\frac{d^2 y}{dx^2}$$**: To find the second derivative, first differentiate $$\frac{dy}{dx} = 3t^3$$ with respect to $$t$$: $$\frac{d}{dt}(3t^3) = 9t^2$$ Now, divide this result by $$\frac{dx}{dt} = \frac{1}{t}$$ to get the second derivative: $$\frac{d^2 y}{dx^2} = \frac{9t^2}{\frac{1}{t}} = 9t^3$$ Thus, the second derivative is: $$y'' = 9t^3$$ If you'd like to express this in terms of $$x$$, recall that $$t = e^x$$ since $$x = \ln t$$. Substituting $$t = e^x$$ into the expression for $$y''$$: $$y'' = 9(e^x)^3 = 9e^{3x}$$ ### Final Answer: $$y'' = 9e^{3x}$$ Let me know if you need further clarification or details. Here are some related questions: 1. How do parametric equations work in general? 2. How is the chain rule used in parametric differentiation? 3. Can you find the curvature of a curve using second derivatives? 4. What are the applications of parametric equations in physics? 5. How can implicit differentiation be used for non-parametric forms? **Tip:** In parametric differentiation, always differentiate both the numerator and denominator with respect to the parameter before dividing.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Parametric Equations
Derivatives
Chain Rule
Formulas
dy/dx = (dy/dt) / (dx/dt)
Second derivative formula d^2y/dx^2
Theorems
Chain Rule
Implicit Differentiation
Suitable Grade Level
Undergraduate Mathematics or Calculus
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