Math Problem Statement
implicit differentiation of: x^8y^9-y=x
Solution
To perform implicit differentiation on the equation:
we will differentiate both sides with respect to , treating as a function of (i.e., ).
Step-by-Step Solution:
- Differentiate the left-hand side:
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For the term , we apply the product rule:
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For the term , we differentiate with respect to , treating as a function of :
Thus, the derivative of the left-hand side is:
- Differentiate the right-hand side:
Now, we have the equation:
- Solve for :
Group all the terms involving :
Factor out on the left-hand side:
Finally, solve for :
Thus, the derivative is:
Would you like further details or have any questions about the steps?
Here are 5 related questions to expand your understanding:
- How would you apply implicit differentiation to ?
- What is the significance of using the product rule when implicitly differentiating products of and ?
- Can you find the second derivative of with respect to for the original equation?
- What happens to the solution if and in the given equation?
- How does implicit differentiation differ from explicit differentiation?
Tip: When implicitly differentiating, always remember to apply the chain rule to the terms involving , since is treated as a function of .
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Product Rule
Chain Rule
Formulas
Product Rule: d(uv)/dx = u'(v) + u(v')
Chain Rule: d(f(g(x)))/dx = f'(g(x)) * g'(x)
Theorems
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Suitable Grade Level
Grades 11-12 or Early College
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