Question

5 points) Consider z=f(x,y) where x=g(t) and y=h(t). Use the Chain Rule to find dz/dt​(3) given that g(3)=2g′(3)=2fx​(2,7)=6​h(3)=7h′(3)=−4fy​(2,7)=−8​

Answer 1

To find dz/dt(3), we need to use the Chain Rule. Substituting the given values into the Chain Rule formula, we find dz/dt(3) = 44.

Step-by-step explanation:

To find dz/dt(3), we need to use the Chain Rule. The Chain Rule states that if we have a function z=f(x,y) where x=g(t) and y=h(t), then dz/dt = dz/dx * dx/dt + dz/dy * dy/dt. In this case, we are given x=g(t), y=h(t), and the partial derivatives fx(2,7), fy(2,7), g(3), g'(3), h(3), and h'(3). We can substitute these values into the Chain Rule formula to find dz/dt(3).

Using the information given, we have:

1. d(z)/dx = fx(2,7) = 6
2. d(z)/dy = fy(2,7) = -8
3. dx/dt = g'(3) = 2
4. dy/dt = h'(3) = -4
5. x= g(3) = 2
6. y = h(3) = 7

Substituting these values into the Chain Rule formula, we get:

dz/dt = dz/dx * dx/dt + dz/dy * dy/dt = (6 * 2) + (-8 * -4) = 12 + 32 = 44

Therefore, dz/dt(3) = 44.