Math Problem Statement
- Let f (x, y) = xyxy sin(xy) + x3y+x4 y2+xy+1 arctan( ln(x) x5+y6 ). Use the limit definition of ∂f ∂y to find ∂f ∂y (1, π). Note: Do not try to find closed form formula for ∂
Solution
To find the partial derivative of with respect to at the point , using the limit definition of the derivative, we'll use:
Given:
- Function:
- Point:
We want to find:
Step-by-Step Computation
Step 1: Write the Limit Definition at
Step 2: Substitute the Function
We will evaluate the function first. This means substituting into the function:
Simplify each term:
- Since , the term involving is zero.
Thus:
Step 3: Apply the Limit Definition
We need to compute:
Substituting the simplified form of :
Now expand:
Thus:
Similarly, evaluate :
Since :
Step 4: Find the Difference
Simplify:
Step 5: Divide by and Take the Limit
Split into individual terms:
Simplify each term:
Take the limit as :
- The other terms become
Thus:
Final Answer
Would you like more details or do you have any questions?
Here are some related questions to expand your understanding:
- How is the limit definition of a partial derivative different from the regular derivative?
- What role does the sine function play in determining the complexity of the limit evaluation?
- How can we generalize this method to calculate higher-order partial derivatives?
- Can you explain why we set early in the process instead of differentiating with respect to first?
- How do trigonometric properties affect the computation of partial derivatives in general?
Tip: When applying the limit definition for partial derivatives, make sure to carefully simplify the expressions to avoid algebraic errors, especially when trigonometric or logarithmic functions are involved.
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Math Problem Analysis
Mathematical Concepts
Partial Derivatives
Multivariable Calculus
Trigonometric Functions
Limit Definition
Formulas
∂f/∂y(x_0, y_0) = lim(h→0) [(f(x_0, y_0 + h) - f(x_0, y_0)) / h]
f(x, y) = xy sin(xy) + x^3y + x^4y^2 + xy + arctan(ln(x) / (x^5 + y^6))
Theorems
Limit Definition of Partial Derivatives
Trigonometric Limits
Suitable Grade Level
College-Level Calculus
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