Math Problem Statement
Given f(x,y) = 3x sin(2xy). Find fx(3,3). Give you answer correct to two decimal places.
Solution
To find , we need to compute the partial derivative of the function with respect to .
Step 1: Compute the partial derivative with respect to
The function is: To find , we differentiate with respect to , treating as a constant.
Using the product rule , we differentiate: Let:
Then:
Next, we differentiate with respect to : [ \frac{\partial}{\partial x} [\sin(2xy)] = \cos(2xy) \cdot \frac{\partial}{\partial x} [2xy] = \cos(2xy) \cdot 2y ]
So:
Step 2: Evaluate at the point
Now, substitute and into :
Simplify the argument of the trigonometric functions:
Step 3: Calculate the values of and
Using the approximate values:
Substitute these values:
Now, calculate:
Final Answer
Would you like further details or have any questions? Here are 5 related questions:
- How would you find for the given function?
- What is the role of the product rule in partial differentiation?
- How does the function behave when ?
- Can you generalize the result to for any point ?
- What is the significance of the angle in trigonometric calculations?
Tip: When computing partial derivatives, carefully apply the product rule, especially in functions involving both multiplication and trigonometric terms.
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Math Problem Analysis
Mathematical Concepts
Partial Differentiation
Product Rule
Trigonometric Functions
Formulas
Partial derivative formula
Product rule for differentiation
Theorems
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Suitable Grade Level
Advanced College Level
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