Math Problem Statement
4x³+y-6xy=0 find the y prime
Solution
To find the derivative of with respect to (denoted as or ), we need to implicitly differentiate the given equation.
The equation is:
Step 1: Differentiate both sides with respect to
We will differentiate each term individually with respect to , keeping in mind that is a function of , so we must apply the chain rule where necessary.
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Differentiate :
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Differentiate with respect to :
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Differentiate with respect to . This requires the product rule:
= -6 \left( x \cdot y' + y \cdot 1 \right) = -6xy' - 6y$$
Step 2: Substitute the derivatives back into the equation
Substitute the differentiated terms back into the original equation:
Step 3: Solve for
To solve for , first combine like terms:
Factor out on the left side:
Finally, solve for :
So the derivative of with respect to is:
Would you like more details or have any questions? Here are 5 questions related to this problem:
- How does the product rule apply when differentiating ?
- What is the significance of implicit differentiation in this problem?
- Can you find the second derivative from this result?
- How does the chain rule help in differentiating terms like with respect to ?
- What are the possible critical points for the function based on this derivative?
Tip: Always remember to apply the chain rule when differentiating a function that depends on another variable.
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Product Rule
Chain Rule
Formulas
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Theorems
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Suitable Grade Level
Advanced High School
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