Math Problem Statement
Solution
We are asked to find the second derivative in terms of and , given the equation .
Step 1: Implicit Differentiation for the First Derivative
We start by differentiating both sides of the equation with respect to , using implicit differentiation.
Using the chain rule for , we get:
Solving for :
Step 2: Implicit Differentiation for the Second Derivative
Now, to find , we differentiate again with respect to :
This requires using the quotient rule:
Here, and . Now, compute the derivatives:
- , and we substitute
Thus:
Now, apply the quotient rule:
Simplify the numerator:
Final Expression:
This is the second derivative of with respect to in terms of and .
Would you like further details on any steps, or have any questions?
Related questions:
- How is implicit differentiation applied in multi-variable calculus?
- What are the advantages of using the quotient rule in differentiation?
- Can you explain how the chain rule works for implicit differentiation?
- How would the result change if the equation was instead?
- What physical phenomena might be modeled by an equation like ?
Tip:
Always simplify the expression as much as possible after differentiating to make interpretation and further manipulation easier.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Chain Rule
Quotient Rule
Formulas
Implicit Differentiation: d/dx (f(y)) = f'(y) * dy/dx
Quotient Rule: d/dx (u/v) = (v * du/dx - u * dv/dx) / v²
Theorems
Chain Rule
Quotient Rule
Suitable Grade Level
University Level Calculus
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