Find fyx for the function below.

Question

Find fyx for the function below.

 

z=f(x,y)=3xy−x^2+5y^3

Answer 1

Since it's difficult to write the notation here, I'm guessing that you mean the second partial derivative of f, first with respect to y, then with respect to x. That's a two-step process.

 

To find the partial derivative wrt y, we treat x as if it were a constant, and differentiate wrt y:

 

f_y(x,y) = 3x + 0 - 15y²

 

Let's see why, before we move on.  Since x is treated as a constant, 3x is the coefficient of y. The derivative of a coefficient times y is just the coefficient, which is 3x.

Since x² is being treated as a constant, its derivative is 0.

I'm sure you understand the 15y²  without any explanation.

 

So, f_y(x,y) = 3x - 15y²

 

Now we take the partial derivative of this wrt x. Remember that we are now treating y as a constant, so 15y² is a constant whose derivative is 0.

 

 

f_yx(x,y) = 3 - 0 = 3.  That's the answer.