Math Problem Statement
Using implicit differentiation, (Second Derivative), Knowing that now, solve If xy-y= 4x then find the 2nd derivative at the point (3,6) in simplest form. Derivate with respect to x, use chain rule for y terms, rearrange terms with y prime on the left, other terms on the right, then divide, simplify any common factors, convert y prime to dy/dx, then find the 2nd derivative using the product or quotient rule, then plug in y, y prime, and x. You neep to apply the point to the 2nd derivative NOT the first
Solution
To find the second derivative of the equation at the point using implicit differentiation, we follow these steps:
Step 1: Implicit Differentiation
Start with the equation: Differentiating both sides with respect to :
Using the product rule on : Rearranging terms gives: Now isolate :
Step 2: Convert to
This can be rewritten as:
Step 3: Find the First Derivative at Point (3, 6)
Substituting and :
Step 4: Second Derivative
Now we differentiate again using the quotient rule: Substituting into the expression: At and :
Final Result
The second derivative at the point is:
Summary and Questions
If you have any further questions or need more details, please let me know! Here are some related questions you might consider:
- How would you find the second derivative using a different implicit function?
- Can you explain how the product rule works in implicit differentiation?
- What are the applications of second derivatives in real-world problems?
- How do you interpret the significance of the second derivative at a point?
- What would happen if the point were different, say ?
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Math Problem Analysis
Mathematical Concepts
Implicit Differentiation
Second Derivatives
Differential Calculus
Formulas
dy/dx = (4 - y) / (x - 1)
d^2y/dx^2 = [(x - 1)(dy/dx) - (4 - y)] / (x - 1)^2
Theorems
Product Rule
Quotient Rule
Suitable Grade Level
Grades 11-12
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