Question

57. Find the total derivative dz/dt, given

(a) z = x^2− 8xy − y^3 , where x = 3t and y = 1 − t.

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

Answer 1

a) z = x^2− 8xy − y^3 , where x = 3t and y = 1 − t

The formula for total derivative is given by,

dz / dt = z_x*dx/dt + z_y*dy/dt, where z_x = dz / dx and z_y = dz / dy.

So,

z_x = 2x - 8y and z_y = - 8x - 3y²

dz/dt = (2x - 8y)*(3) + ( - 8x - 3y²)*(-1) = (2x - 8y)*(3) + (8x + 3y²)*(1)

dz / dt = (2*3t - 8(1 - t))*(3) + (8*3 - 3(1 - t)²)

dz/dt = (6t - 8 + 8t)*3 + 24 - 3(1 - 2t + t²)

dz/dt = 42t - 24 + 24 - 3 + 6t - 3t²

dz/dt = 48t - 3t² - 3

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

dz / dt = (dz / dx)(dx/dt) + (dz/dy)(dy/dt) + (dz/dt)(dt/dt)

dz / dt = z_x(b) + z_y(0) + z_t(1)

dz/dt = b*z_x + z_t