MHB Can someone help me solve these problems?

[khokababu]

1. If z=f(x,y)=x3+xey and x=sint,y=logt, use the chain rule to find dzdt in terms of t. Then - by using direct substitution:

express z in terms of t and find dzdt .

2. Suppose that z is a function of x and y, implicitly related by the equation

x2/4+y2+z2/9=3

Find the equation of the tangent plane to the surface f(x,y,z)=3 (surface of ellipsoid) at the point where (x,y)=(−2,1,−3). Then determine the linear approximation to function z in vicinity of (x,y)=(−2,1) and use it to find the approximate value of z(−2.1,1.1)

 

[HOI]

A couple of tips:
1) use ^ to indicate powers I think the first function is x^3+ xe^y but cannot be certain.

2) Show some effort yourself, don't just post a problem without showing any of our own ideas or work.

Can you find the partial derivatives of x^3+ xe^y with respect to x and y? Since the first problem says "use the chain rule" do you know what that is?

For the second problem, you have f(x, y, z)= x^2/4+ y^2+ z^2/9= 3. If you have a surface given by f(x, y, z)= 3, then the tangent plane at (x_0, y_0, z_) (which satisfy the equation f(x_0, y_0, z_0)= 3) then the tangent plane there is given by \frac{\partial f}{\partial x}(x- x_0)+ \frac{\partial f}{\partial y}(y- y_0)+ \frac{\partial f}{\partial z}(z- z_0)= 0 where the partial derivatives are evaluated at (x_0, y_0, z_0). Can you find those partial derivatives?