Math Problem Statement
when finding an wuation that describes the relationship of rates of change of x and y, for (x^2)+3y=4, why do we multiply 2x by dx/dt
Solution
When differentiating an equation like to find the relationship between the rates of change of and , we use implicit differentiation with respect to time . Here's why is multiplied by :
1. Implicit Differentiation Explanation
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The equation involves two variables, and , that may be functions of time . Therefore, we differentiate both sides of the equation with respect to , treating and as functions of .
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For the term , we apply the chain rule: The reasoning is that depends on , and depends on . So, when we differentiate , we not only compute the derivative but also multiply by the rate of change of with respect to , which is .
2. Why Multiply by ?
- If were a constant, its derivative would be , and there would be no .
- However, since changes with , the chain rule ensures that any derivative involving must account for how itself varies over time.
Step-by-Step Differentiation of :
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Differentiate :
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Differentiate (using the chain rule):
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Differentiate the constant :
3. Combine Results:
This equation relates and , the rates of change of and with respect to time.
Key Idea:
The multiplication by comes from the chain rule, which accounts for the dependency of on .
Would you like to see further examples or clarification? Here are five related questions you might consider exploring:
- How does the chain rule apply to more complex functions of multiple variables?
- What is the physical significance of rates of change ( and ) in real-world contexts?
- How would this differentiation look if we had ?
- Can implicit differentiation help solve for specific or values?
- What happens if or is a constant during differentiation?
Tip: Always think of implicit differentiation as "tracking dependencies" between variables as they change over time!
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Rates of Change
Chain Rule
Formulas
d/dt(x^2) = 2x * dx/dt
d/dt(3y) = 3 * dy/dt
2x * dx/dt + 3 * dy/dt = 0
Theorems
Chain Rule of Differentiation
Suitable Grade Level
Grades 10-12
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