Find the second order partial derivatives

Question

f(x, y)=  x^5 + x^2y + x + 1

Answer 1

There are three second order partial derivatives with a 2 variable function:

f_xx, f_yy and f_xy=f_yx

 

First, find the first order derivatives

 

∂/∂x f(x,y) = f_x(x,y) = 5x⁴ + 2xy + 1

∂/∂y f(x,y) = f_y(x,y) = x²

 

Now take another derivative of each, with respect to each variable

∂/∂x f_x(x,y) = f_xx(x,y) = 20x³ + 2y

∂/∂y f_x(x,y) = f_xy(x,y) = 2x

 

∂/∂x f_y(x,y) = f_yx(x,y) = 2x

∂/∂y f_y(x,y) = f_yy(x,y) = 0

 

Note that the mixed partial derivative is the same regardless of which order you take the derivatives in.

Answer 2

f(x,y) = x⁵ + x²y + x + 1

 

 

First, the 1st partial derivatives:

 

f_x = 5x⁴ + 2xy + 1

 

f_y = x²

 

 

Now, the 2nd partial derivatives:

 

f_xx = 20x³ + 2y

 

f_yy = 0

 

f_xy = f_yx = 2x