Math Problem Statement
Given f(x,y) = 4x sin(5xy). Find fx(5,3). Give you answer correct to two decimal places.
Solution
To find , we first 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(5xy)] = \cos(5xy) \cdot \frac{\partial}{\partial x} [5xy] = \cos(5xy) \cdot 5y ]
So: Simplifying further:
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 geometric interpretation of at a specific point?
- How does the chain rule apply to partial derivatives?
- What changes when computing if is a more complex function?
- Can you find the critical points of the function ?
Tip: Always check the angle mode (degrees vs radians) when evaluating trigonometric functions, especially when specific angle values are given.
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Math Problem Analysis
Mathematical Concepts
Partial Derivatives
Product Rule
Trigonometric Functions
Formulas
Product rule for partial derivatives
Theorems
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Suitable Grade Level
Advanced Undergraduate
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