f(xy)= x^2e^3xy find fx fy (1,-1)
fy (3,-3)
To find the partial derivatives fx and fy, we need to apply the chain rule. The chain rule states that if we have a function f(u) where u is a function of variables x and y (in this case, u = xy), then the partial derivative of f with respect to x is given by (df/du) * (du/dx), and the partial derivative of f with respect to y is given by (df/du) * (du/dy).
Let's start by finding fx (the partial derivative of f with respect to x):
1. Start with the function f(u) = x^2 * e^(3xy).
2. Replace u with xy to get f(xy) = (xy)^2 * e^(3xy).
3. Differentiate f(xy) with respect to x, treating y as a constant. This gives us df/dx.
4. Apply the chain rule: (df/dx) = (df/du) * (du/dx).
5. Differentiate f(u) = u^2 * e^(3u) with respect to u: (df/du) = 2u * e^(3u).
6. Differentiate u = xy with respect to x: (du/dx) = y.
Now we can multiply the two derivatives together to get fx:
fx = (df/du) * (du/dx)
= (2u * e^(3u)) * y
= 2(xy) * e^(3xy) * y.
To find fy (the partial derivative of f with respect to y), we follow the same steps as above, but differentiate with respect to y instead of x:
1. Start with the function f(u) = x^2 * e^(3xy).
2. Replace u with xy to get f(xy) = (xy)^2 * e^(3xy).
3. Differentiate f(xy) with respect to y, treating x as a constant. This gives us df/dy.
4. Apply the chain rule: (df/dy) = (df/du) * (du/dy).
5. Differentiate f(u) = u^2 * e^(3u) with respect to u: (df/du) = 2u * e^(3u).
6. Differentiate u = xy with respect to y: (du/dy) = x.
Multiply the two derivatives together to get fy:
fy = (df/du) * (du/dy)
= (2u * e^(3u)) * x
= 2(xy) * e^(3xy) * x.
Now let's find fx and fy at the given points (1,-1) and (3,-3):
For point (1,-1):
Substitute x = 1 and y = -1 into the formulas we derived above:
fx(1,-1) = 2(1*(-1)) * e^(3*1*(-1)) * (-1)
= 2 * (-1) * e^(-3)
= -2e^(-3).
fy(1,-1) = 2(1*(-1)) * e^(3*1*(-1)) * 1
= 2 * (-1) * e^(-3)
= -2e^(-3).
For point (3,-3):
Substitute x = 3 and y = -3 into the formulas we derived above:
fx(3,-3) = 2(3*(-3)) * e ^ (3*3*(-3)) * (-3)
= 2 * (-9) * e^(-27)
= -18e^(-27).
fy(3,-3) = 2(3*(-3)) * e^(3*3*(-3)) * 3
= 2 * (-9) * e^(-27)
= -18e^(-27).
Therefore, the values of fx and fy at the points (1,-1) and (3,-3) are as follows:
fx(1,-1) = fy(1,-1) = -2e^(-3),
fx(3,-3) = fy(3,-3) = -18e^(-27).