Math Problem Statement
use implicit differentiation to find dy/dx of 5x^3=-3xy+2 in terms of y
Solution
To find using implicit differentiation for the equation:
Step 1: Differentiate both sides with respect to .
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Differentiate the left side:
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Differentiate the right side: Apply the product rule to the term and differentiate , which is constant:
- The derivative of using the product rule is:
- The derivative of the constant is .
So, the differentiated equation becomes:
Step 2: Solve for .
Now, isolate : Add to both sides: Now, divide both sides by :
Simplify:
Finally, factor out the common factor of 3:
Final Answer:
Would you like more details on any part of this process? Here's a list of related questions to further explore:
- What are the product and chain rules used for implicit differentiation?
- How do you apply implicit differentiation to more complex equations?
- What happens if the equation includes higher-order terms like or trigonometric functions?
- How do you check if an implicit derivative is correct after differentiating?
- What is the geometric interpretation of implicit differentiation in terms of slopes?
Tip: Remember that in implicit differentiation, when differentiating terms involving , treat as a function of , meaning you apply the chain rule ().
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Product Rule
Algebraic Manipulation
Formulas
Implicit differentiation: d/dx[f(y)] = f'(y) dy/dx
Product rule: d/dx[uv] = u'v + uv'
Theorems
Chain Rule
Suitable Grade Level
Grades 11-12