This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*
**Unformatted text preview: **15.5 The Chain Rule Warm-Up Problem: The temperature (in ◦ C) at a point ( x, y ) on a metal plate in the xy-plane is T ( x, y ) = x 3 + 2 y 2 + x. Assume that distance is measured in centimeters. Find the rate at which temperature changes with respect to distance if we start at the point (1 , 2) and move 1. to the right and parallel to the x-axis 2. upward and parallel to the y-axis. Chain Rule (Version 1): Let z = f ( x, y ) be a differentiable function of x and y , with x = x ( t ) and y = y ( t ), where x and y are differentiable functions of t . Then z is a differentiable function of t and dz dt = ∂f ∂x dx dt + ∂f ∂y dy dt . Proof: Since f is a differentiable function of x and y , we know that there exist functions 1 and 2 such that Δ z = f x Δ x + f y Δ y + 1 Δ x + 2 Δ y, with lim (Δ x, Δ y ) → (0 , 0) 1 = 0 = lim (Δ x, Δ y ) → (0 , 0) 2 . Thus we have dz dt = lim Δ t → ∂f ∂x · Δ x + ∂f ∂y · Δ y + 1 Δ x + 2 Δ y Δ t = lim Δ t → ∂f ∂x...

View
Full
Document