The Chain Rule and Second Partials -- Answers

Question 1

If w = w(x,y), x = sr, y = s²r³, find w_r.

Answer: First notice, x_r = s, and y_r = 3s²r². From the chain rule we know that w_r = w_xx_r + w_yy_r. Therefore, w_r = sw_x + 3s²r²w_y.

When you understand this solution, click here to return to the discussion on the chain rule and second partial derivatives.

Question 2

Let h(x,y) = ye^-xy. Find h_y.

Answer: We want h_y. So, we differentiate with respect to y and treat x as a constant. Using the product rule, we get:

h_y = e^-xy - xye^-xy.

When you understand this solution, click here to return to the discussion on the chain rule and second partial derivatives.

Question 3

Let h(x,y) = ye^-xy. Find h_yy.

Answer: From our work above, h_y = e^-xy - xye^-xy. Now, we take the partial derivative of h_y. We find:

h_yy = -2xe^-xy + x²ye^-xy.

When you understand this solution, click here to return to the discussion on the chain rule and second partial derivatives.

Question 4

Find w_sr where w = w(x,y), x = s + 2r, and y = s³ - rs.

Answer: First, w_s = w_xx_s + w_yy_s = w_x + (3s² - r)w_y. Next, w_sr = w_xr - w_y + (3s² - r)w_yr. We need to expand w_xr and w_yr. Using the Chain Rule, we find w_xr = w_xxx_r + w_xyy_r = 2w_xx - w_xys. Similarly, w_yr = 2w_yx - w_yys. Substituting back into our equation, we find w_sr = 2w_xx - w_xys - w_y + (3s² - r)(2w_yx - w_yys).

When you understand this solution, click here to return to the discussion on the chain rule and second partial derivatives.

Last modified: March, 1997
Created by Tim Chartier