Find the first partial derivative of function. z=xsin(xy)
find dz/dx and dz/dy.
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
can you show it with using product rule
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)
To find the first partial derivatives of the function z = xsin(xy) with respect to x and y, we will differentiate the function with respect to each variable separately while treating the other variable as a constant.
1. Finding dz/dx:
To find the partial derivative dz/dx, we differentiate the function with respect to x while treating y as a constant.
Step 1: Differentiate the function z = xsin(xy) with respect to x:
d/dx(xsin(xy))
Step 2: Apply the product rule to differentiate the function:
= x * d/dx(sin(xy)) + sin(xy) * d/dx(x)
Step 3: Differentiate the terms separately:
= x * (cos(xy) * d/dx(xy)) + sin(xy) * 1
Step 4: Simplify:
= x * (cos(xy) * (y * d/dx(x) + x * d/dx(y))) + sin(xy)
Note: The derivative of xy with respect to x is y, and the derivative of x with respect to x is 1.
Simplifying further, we get:
= x * (cos(xy) * (y * 1 + x * 0)) + sin(xy)
= xycos(xy) + sin(xy)
Therefore, the partial derivative dz/dx is xycos(xy) + sin(xy).
2. Finding dz/dy:
To find the partial derivative dz/dy, we differentiate the function with respect to y while treating x as a constant.
Step 1: Differentiate the function z = xsin(xy) with respect to y:
d/dy(xsin(xy))
Step 2: Apply the product rule to differentiate the function:
= x * d/dy(sin(xy)) + sin(xy) * d/dy(x)
Step 3: Differentiate the terms separately:
= x * (cos(xy) * d/dy(xy)) + sin(xy) * 0
Step 4: Simplify:
= x * (cos(xy) * (x * d/dy(y) + y * d/dy(x))) + 0
Note: The derivative of xy with respect to y is x, and the derivative of x with respect to y is 0.
Simplifying further, we get:
= x * (cos(xy) * (x * 0 + y * 1))
= xycos(xy)
Therefore, the partial derivative dz/dy is xycos(xy).
In summary, dz/dx is xycos(xy) + sin(xy), and dz/dy is xycos(xy).