Math Problem Statement
Solution
To solve the problem using implicit differentiation, we are given the equation:
We need to find . To achieve this, we will differentiate both sides of the equation with respect to , treating as a function of .
Steps for implicit differentiation:
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Differentiate with respect to : Since is not a function of , we treat it as a constant.
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Differentiate with respect to :
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Differentiate with respect to : Here, we need to use the product rule:
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Differentiate with respect to : Again, is not a function of , so it is treated as a constant.
Now, putting everything together:
Now, we solve for :
Thus, the derivative is:
This is the expression for .
Would you like further details or clarifications? Here are five related questions:
- How is implicit differentiation different from explicit differentiation?
- Can this method be extended to higher-order derivatives?
- What is the significance of treating some variables as constants in implicit differentiation?
- How does the product rule apply in cases like ?
- Can implicit differentiation be used for non-polynomial functions?
Tip: Always remember to apply the chain rule when differentiating implicitly, especially when dealing with products of variables.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Multivariable Calculus
Product Rule
Formulas
∂/∂z (x^6w) = x^6 ∂w/∂z
∂/∂z (w^7) = 7w^6 ∂w/∂z
∂/∂z (wz^2) = z^2 ∂w/∂z + 2wz
∂/∂z (5yz) = 5y
Theorems
Chain Rule
Product Rule
Suitable Grade Level
Undergraduate Level (Calculus)
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