Math Problem Statement
Find ππ¦ππ₯οΏ½οΏ½οΏ½οΏ½ by implicit differentiation.
2π₯^3π¦^2β4π₯^2π¦=4
Solution
To find by implicit differentiation, we'll differentiate both sides of the equation with respect to , while treating as a function of (i.e., using the chain rule when differentiating terms involving ).
Step 1: Differentiate both sides
We differentiate each term:
First term:
Using the product rule for differentiation: , where and , we get:
Simplified:
Second term:
Again, apply the product rule: and , so:
Right-hand side:
Step 2: Combine results
Putting everything together, we have:
Step 3: Solve for
Group the terms involving on one side:
Factor out on the left side:
Finally, solve for :
Final Answer:
Would you like more details on any step, or have any questions?
Here are 5 related questions you might be interested in:
- How does implicit differentiation work?
- What is the product rule in differentiation?
- How does the chain rule apply when differentiating implicitly?
- Can implicit differentiation be applied to trigonometric functions?
- How do you solve implicit equations after differentiation?
Tip: When using implicit differentiation, always remember to apply the product rule and chain rule carefully to terms involving both and .
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Product Rule
Chain Rule
Formulas
Product Rule: (uv)' = u'v + uv'
Chain Rule: dy/dx = dy/du * du/dx
Theorems
Implicit Differentiation Theorem
Suitable Grade Level
Grades 11-12 (Advanced High School Calculus)