Question

Let f(x, y, z) = x + ln(2y + 3z 2 ). Compute ∂f ∂x, ∂ 2f ∂y∂x, and ∂ 3f ∂z∂y∂x. Evaluate ∂ 2f ∂y∂x(2, 1, −2).

Answer 1

Explanation:

Given that

`f(x, y, z) = x + ln(2y + 3z^2 )`

Let us find partial derivatives one by one

`(∂f )/(∂x) =1\\(∂^2f )/(∂y∂x) =(∂ )/(∂y)(1) =0\\(∂^3f )/(∂z∂y∂x) =(∂ )/(∂z)(0) =0`

At the point (2,1,-2)

`(∂^2f )/(∂y∂x)=0`

(since at all points the value is constant 0)