I need help with this problem?

[Power]

If f(x, y, z)=sin(3x-yz), where x=e^(t-1), y=t^3, z=t-2, what's df/dt(1)?

df/dt=(df/dx)(dx/dt)+(df/dy)(dy/dt)
+(df/dz)(dz/dt)
=(3cos(3x-yz))(e^(t-1))+
(cos(3x-z))(3t^2)+
(cos(3x-y))(1)
=(3cos(3x-yz))(e^0)+
(cos(3x-z))(3)+
(cos(3x-y))
=(3cos(3x-yz))+
(cos(3x-z))(3)+
(cos(3x-y))

What should I do next? Show your work through steps. Thanks.

 

[DrPhil]

If f(x, y, z) = sin(3x - yz),
where x = e^(t-1), y = t^3, z = t-2, what's df/dt(1)?

df/dt = (df/dx)(dx/dt) + (df/dy)(dy/dt) + (df/dz)(dz/dt)
.......= cos(3x - yz) (3) (e^(t-1))
........+ cos(3x - yz) (-z) (3t^2)
........+ cos(3x - yz )(-y) (1)
= . . .

What should I do next? Show your work through steps. Thanks.

The partials of f with respect to x, y, and z all require the chain rule, with the first factor being the derivative of the sine - the same for all three coordinates. The second factor is the coefficient of x, y, and z respectively within the argument of the sine, (3x - yz): the coefficient of x is 3, of y is -z, and of z is -y. I have corrected your work above.

Once you get back to the same point you were, you need to evaluate at t=1:
x(1) = .. , y(1) = .. , z(1) = ..
(3x - yz) = ..
. . .