Math Problem Statement
y = √ e^−2t^2 + 4 ,dy/dt =
Solution
To find , we need to use the chain rule. The given function is:
Step 1: Apply the chain rule
The derivative of with respect to is:
Here, . Let's find .
Step 2: Differentiate with respect to
The derivative of with respect to is found using the chain rule:
[ \frac{d}{dt}[e^{-2t^2}] = e^{-2t^2} \cdot \frac{d}{dt}[-2t^2] = e^{-2t^2} \cdot (-4t) ]
The derivative of the constant is . Thus:
Step 3: Combine results
Now substitute and back into the chain rule formula:
Simplify:
Final Answer:
Let me know if you'd like further clarification or details on any step!
Relative Questions:
- How does the chain rule apply in similar cases involving square roots?
- What is the derivative of for any constants and ?
- How would the answer change if the square root was replaced by a cube root?
- What is the significance of the term in determining the behavior of ?
- How do we handle higher-order derivatives for this function?
Tip: Always simplify intermediate results before substituting back to avoid unnecessary complexity!
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Math Problem Analysis
Mathematical Concepts
Differentiation
Chain Rule
Exponential Functions
Square Root Functions
Formulas
dy/dt = (1 / 2√u) * (du/dt)
d/dt[e^(-kt^n)] = e^(-kt^n) * (-k * n * t^(n-1))
Theorems
Chain Rule
Suitable Grade Level
Grades 11-12