Find the total derivative dz/dt, given
z=f(x,y,t) where x=a+bt and y=c+dt
To find the total derivative dz/dt, we need to use the chain rule of differentiation. The chain rule states that if z is a function of x, y, and t, and if x and y are functions of t, then the total derivative dz/dt is given by:
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) + (∂z/∂t)
Let's break down the steps to find dz/dt:
1. Given that x = a + bt and y = c + dt, we have the following expressions for dx/dt and dy/dt:
dx/dt = b
dy/dt = d
2. Differentiate z = f(x, y, t) with respect to x, y, and t separately to find (∂z/∂x), (∂z/∂y), and (∂z/∂t), respectively.
3. Substitute the values obtained in step 2 into the chain rule equation:
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) + (∂z/∂t)
This will give you the total derivative dz/dt.