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 fx(5,3)f_x(5,3), we first need to compute the partial derivative of the function f(x,y)=4xsin(5xy)f(x, y) = 4x \sin(5xy) with respect to xx.

Step 1: Compute the partial derivative with respect to xx

The function is: f(x,y)=4xsin(5xy)f(x, y) = 4x \sin(5xy) To find fx(x,y)f_x(x, y), we differentiate f(x,y)f(x, y) with respect to xx, treating yy as a constant.

Using the product rule, ddx[u(x)v(x)]=u(x)v(x)+u(x)v(x)\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x), we differentiate: fx(x,y)=x[4xsin(5xy)]f_x(x, y) = \frac{\partial}{\partial x} [4x \sin(5xy)] Let:

  • u(x)=4xu(x) = 4x
  • v(x)=sin(5xy)v(x) = \sin(5xy)

Then: fx(x,y)=4sin(5xy)+4xx[sin(5xy)]f_x(x, y) = 4 \sin(5xy) + 4x \cdot \frac{\partial}{\partial x} [\sin(5xy)]

Next, we differentiate sin(5xy)\sin(5xy) with respect to xx: [ \frac{\partial}{\partial x} [\sin(5xy)] = \cos(5xy) \cdot \frac{\partial}{\partial x} [5xy] = \cos(5xy) \cdot 5y ]

So: fx(x,y)=4sin(5xy)+4x5ycos(5xy)f_x(x, y) = 4 \sin(5xy) + 4x \cdot 5y \cos(5xy) Simplifying further: fx(x,y)=4sin(5xy)+20xycos(5xy)f_x(x, y) = 4 \sin(5xy) + 20xy \cos(5xy)

Step 2: Evaluate at the point (x,y)=(5,3)(x, y) = (5, 3)

Now, substitute x=5x = 5 and y=3y = 3 into fx(x,y)f_x(x, y): fx(5,3)=4sin(5×5×3)+20×5×3cos(5×5×3)f_x(5, 3) = 4 \sin(5 \times 5 \times 3) + 20 \times 5 \times 3 \cos(5 \times 5 \times 3)

Simplify the argument of the trigonometric functions: fx(5,3)=4sin(75)+300cos(75)f_x(5, 3) = 4 \sin(75) + 300 \cos(75)

Step 3: Calculate the values of sin(75)\sin(75^\circ) and cos(75)\cos(75^\circ)

Using the approximate values: sin(75)0.9659\sin(75^\circ) \approx 0.9659 cos(75)0.2588\cos(75^\circ) \approx 0.2588

Substitute these values: fx(5,3)=4×0.9659+300×0.2588f_x(5, 3) = 4 \times 0.9659 + 300 \times 0.2588

Now, calculate: fx(5,3)=3.8636+77.64=81.50f_x(5, 3) = 3.8636 + 77.64 = 81.50

Final Answer

fx(5,3)81.50f_x(5, 3) \approx \boxed{81.50}

Would you like further details or have any questions? Here are 5 related questions:

  1. How would you find fy(5,3)f_y(5, 3) for the given function?
  2. What is the geometric interpretation of fx(x,y)f_x(x, y) at a specific point?
  3. How does the chain rule apply to partial derivatives?
  4. What changes when computing fx(x,y)f_x(x, y) if f(x,y)f(x, y) is a more complex function?
  5. Can you find the critical points of the function f(x,y)=4xsin(5xy)f(x, y) = 4x \sin(5xy)?

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