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 fx(3,3)f_x(3,3), we need to compute the partial derivative of the function f(x,y)=3xsin(2xy)f(x, y) = 3x \sin(2xy) with respect to xx.

Step 1: Compute the partial derivative with respect to xx

The function is: f(x,y)=3xsin(2xy)f(x, y) = 3x \sin(2xy) 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[3xsin(2xy)]f_x(x, y) = \frac{\partial}{\partial x} [3x \sin(2xy)] Let:

  • u(x)=3xu(x) = 3x
  • v(x)=sin(2xy)v(x) = \sin(2xy)

Then: fx(x,y)=3sin(2xy)+3xx[sin(2xy)]f_x(x, y) = 3 \sin(2xy) + 3x \cdot \frac{\partial}{\partial x} [\sin(2xy)]

Next, we differentiate sin(2xy)\sin(2xy) with respect to xx: [ \frac{\partial}{\partial x} [\sin(2xy)] = \cos(2xy) \cdot \frac{\partial}{\partial x} [2xy] = \cos(2xy) \cdot 2y ]

So: fx(x,y)=3sin(2xy)+6xycos(2xy)f_x(x, y) = 3 \sin(2xy) + 6xy \cos(2xy)

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

Now, substitute x=3x = 3 and y=3y = 3 into fx(x,y)f_x(x, y): fx(3,3)=3sin(2×3×3)+6×3×3cos(2×3×3)f_x(3, 3) = 3 \sin(2 \times 3 \times 3) + 6 \times 3 \times 3 \cos(2 \times 3 \times 3)

Simplify the argument of the trigonometric functions: fx(3,3)=3sin(18)+54cos(18)f_x(3, 3) = 3 \sin(18) + 54 \cos(18)

Step 3: Calculate the values of sin(18)\sin(18^\circ) and cos(18)\cos(18^\circ)

Using the approximate values: sin(18)0.3090\sin(18^\circ) \approx 0.3090 cos(18)0.9511\cos(18^\circ) \approx 0.9511

Substitute these values: fx(3,3)=3×0.3090+54×0.9511f_x(3, 3) = 3 \times 0.3090 + 54 \times 0.9511

Now, calculate: fx(3,3)=0.927+51.3594=52.29f_x(3, 3) = 0.927 + 51.3594 = 52.29

Final Answer

fx(3,3)52.29f_x(3, 3) \approx \boxed{52.29}

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

  1. How would you find fy(3,3)f_y(3, 3) for the given function?
  2. What is the role of the product rule in partial differentiation?
  3. How does the function f(x,y)=3xsin(2xy)f(x, y) = 3x \sin(2xy) behave when y=0y = 0?
  4. Can you generalize the result to fx(a,b)f_x(a, b) for any point (a,b)(a, b)?
  5. What is the significance of the angle 1818^\circ 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