Answer:
i) See explanations below
ii) partial derivative of w with respect to u = 2u + 4uv
partial derivative of w with respect to v = -4v + 2u²
The partial derivative of w with respect to u = 4/3
The partial derivative of w with respect to v = 8/9
Step-by-step explanation:
w=xy+yz+xz
x=u+2v
y=u-2v
z=uv
First we are to express partial derivative of w with respect to u and v respectively using chain rule.
See 1st attachment.
Inserting the value of x, y and z with respect to u and v in w:
w = (u+2v)(u-2v) + (u-2v)(uv) + (u+2v)(uv)
w = u² -2uv +2uv - 2v² +u²v -2uv² + u²v + 2uv²
w = u² - 2v² + 2u²v
We would find the partial derivative of w with respect to u and v respectively.
Partial derivative is done by differentiating a function with one variable while keeping the other variable constant.
At the point (2/3, 0) given, we get the value for the partial derivative of w with respect to u and v respectively.
See 2nd attachment for more explanations.
