Find the first partial derivative of function. z=xsin(xy)

Question

find dz/dx and dz/dy.

Answer 1

To find the partial derivative with respect to x, we treat y as a constant and apply the product rule:

dz/dx = sin(xy) + xcos(xy) * y

To find the partial derivative with respect to y, we again use the product rule, but this time treat x as a constant:

dz/dy = xcos(xy) * x

Answer 2

can you show it with using product rule

Answer 3

Sure! Here's how we can find both partial derivatives using the product rule:

For dz/dx:

- Let u(x) = x and v(x, y) = sin(xy)
- Then du/dx = 1 and dv/dx = y*cos(xy) (using chain rule)
- Applying the product rule, we get: dz/dx = u * dv/dx + v * du/dx = x*y*cos(xy) + sin(xy)*1
- Simplifying this expression, we get: dz/dx = sin(xy) + xy*cos(xy)

For dz/dy:

- Let u(y) = y and v(x, y) = x*sin(xy)
- Then du/dy = 1 and dv/dy = x*cos(xy) + x*cos(xy)*y (using product and chain rule)
- Applying the product rule, we get: dz/dy = u * dv/dy + v * du/dy = x*cos(xy) + x*y*cos(xy) + x*sin(xy)*1
- Simplifying this expression, we get: dz/dy = x*cos(xy) * (1 + y) + x*sin(xy)