Compute ∂^2f/ ∂y∂x?

Question

Let f(x, y) = e^x^(2+3y) . You are given that ∂f/∂x = 2xe^x^(2+3y)

and Compute ∂^2f/∂x^2

Please provide explanation and steps. Thank you

Answer 1

Let u = x^(3+2y)

Taking the derivative of 2xe^u with respect to y is (2xe^u)(du/dy)

Take the logarithm of both sides of u resulting in

ln(u) = (3+2y)ln(x)

Now take the derivative of both sides:

(1/u)du/dy = 2ln(x) So, du/dy = 2uln(x) = 2ln(x)x^(3+2y)

So, the final answer is

f_xy = (2xe^u)(2uln(x)) with u = x^(3+2y)