Question

Find the total derivative dz/dt, given

(a) z = x²− 8xy WQ− y³ , where x = 3t and y = 1 − t.
(b) z = f(x, y, t), where x = a + bt, and y = c + kt

Answer 1

To find the total derivative dz/dt, you need to find the partial derivatives of z with respect to x and y, and then multiply them by the rates of change of x and y with respect to t. For the given functions, (a) z = x² - 8xy + y³ where x = 3t and y = 1 - t, and (b) z = f(x, y, t) where x = a + bt and y = c + kt.

Step-by-step explanation:

To find the total derivative dz/dt for the given functions, we need to find the partial derivatives of z with respect to x and y, and then multiply them by the rates of change of x and y with respect to t.

(a) For z = x² - 8xy + y³ where x = 3t and y = 1 - t:

∂z/∂x = 2x - 8y

∂z/∂y = -8x + 3y²

dx/dt = 3

dy/dt = -1

Using the chain rule, dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt) = (2x - 8y)(3) + (-8x + 3y²)(-1)

(b) For z = f(x, y, t) where x = a + bt and y = c + kt:

∂z/∂x = ∂f/∂x, ∂z/∂y = ∂f/∂y, dx/dt = b, dy/dt = k

So, dz/dt = (∂f/∂x)(b) + (∂f/∂y)(k)

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