What is the derivative?

Question

Find the following derivatives.

z_s and z_t, where z = e^(5x+3y), x=8st, and y = 2s+9t

z_{s = ?}

z_{t = ?}

Answer 1

The problem is asking for ∂z/∂s = z_s and ∂z/∂t = z_t.

First, need to find ∂x/∂s = x_s, ∂x/∂t = x_t, ∂y/∂s = y_s and ∂y/∂t = y_t.

x_s = 8t

x_t = 8s

y_s = 2

y_t = 9

z_s = e^5x+3y(5x_s + 3y_s) = e^5x+3y(40t + 6) = e^{40st + 6s+27t}(40t + 6)

z_t = e^5x+3y(5x_t + 3y_t) = e^5x+3y(40s + 27) = e^{40st + 6s+27t}(40s + 27)