Given f(x,y) = 4x sin(5xy).  Find fx(5,3). Give you answer correct to two decimal places.

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

### Step 1: Compute the partial derivative with respect to \( x \)
The function is:
\[
f(x, y) = 4x \sin(5xy)
\]
To find \( f_x(x, y) \), we differentiate \( f(x, y) \) with respect to \( x \), treating \( y \) as a constant.

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

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

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

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

### Step 2: Evaluate at the point \( (x, y) = (5, 3) \)
Now, substitute \( x = 5 \) and \( y = 3 \) into \( f_x(x, y) \):
\[
f_x(5, 3) = 4 \sin(5 \times 5 \times 3) + 20 \times 5 \times 3 \cos(5 \times 5 \times 3)
\]

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

### Step 3: Calculate the values of \( \sin(75^\circ) \) and \( \cos(75^\circ) \)
Using the approximate values:
\[
\sin(75^\circ) \approx 0.9659
\]
\[
\cos(75^\circ) \approx 0.2588
\]

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

Now, calculate:
\[
f_x(5, 3) = 3.8636 + 77.64 = 81.50
\]

### Final Answer
\[
f_x(5, 3) \approx \boxed{81.50}
\]

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

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

**Tip:** Always check the angle mode (degrees vs radians) when evaluating trigonometric functions, especially when specific angle values are given.

Learn how to compute the partial derivative f_x(5,3) for the function f(x,y) = 4x sin(5xy). Follow a step-by-step guide using the product rule for partial derivatives and trigonometric function evaluations. Suitable for advanced undergraduate students and professionals.

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