Question

Find the first partial derivatives of the function z = x sin(xy).

1) ∂z/∂x = sin(xy) + x²y cos(xy)
2) ∂z/∂x = sin(xy) + xy cos(xy)
3) ∂z/∂x = sin(xy) + y cos(xy)
4) ∂z/∂x = sin(xy) + x² cos(xy)

Answer 1

The first partial derivative of the function z = x sin(xy) with respect to x is:

∂z/∂x = sin(xy) + xy cos(xy) Option 2 is answer.

Step-by-step explanation:

To find the partial derivative of z with respect to x, we treat y as a constant and then use the product rule and chain rule.

Step 1: Apply the product rule:

∂z/∂x = ∂/∂x [x sin(xy)] = sin(xy) + x ∂/∂x [sin(xy)]

Step 2: Apply the chain rule:

∂/∂x [sin(xy)] = y cos(xy)

Therefore, the complete first partial derivative is:

∂z/∂x = sin(xy) + x * y cos(xy) = sin(xy) + xy cos(xy)

Option 2 is answer.