Find all the first and second order partial derivatives of f(x, y) = cos^3 (3x ^ 2 - y) + log_y(x)

To find the first and second order partial derivatives of f(x, y) = cos^3 (3x ^ 2 - y) + log_y(x), we will first find the partial derivative with respect to x and then with respect to y.

Step 1: Finding the first order partial derivative with respect to x: To find the partial derivative of f(x, y) with respect to x, we differentiate each term of the function with respect to x while treating y as a constant.

The derivative of cos^3 (3x ^ 2 - y) with respect to x can be found using the chain rule. Let's denote the inner function as u = 3x ^ 2 - y. The derivative of cos^3 u with respect to u is -3cos^2 u * sin u. Now, we multiply this by the derivative of u with respect to x, which is 6x. Therefore, the partial derivative of cos^3 (3x ^ 2 - y) with respect to x is -18x * cos^2 (3x ^ 2 - y) * sin (3x ^ 2 - y).

The derivative of log_y(x) with respect to x can be found using the logarithmic differentiation rule. The derivative of log_y(x) with respect to x is (1 / (x * ln(y))).

Step 2: Finding the first order partial derivative with respect to y: To find the partial derivative of f(x, y) with respect to y, we differentiate each term of the function with respect to y while treating x as a constant.

The derivative of cos^3 (3x ^ 2 - y) with respect to y is -cos^3 (3x ^ 2 - y).

The derivative of log_y(x) with respect to y can be found using the logarithmic differentiation rule. The derivative of log_y(x) with respect to y is (-ln(x) / (y * ln(y) ^ 2)).

Step 3: Finding the second order partial derivatives: To find the second order partial derivatives, we differentiate the first order partial derivatives obtained in Step 1 and Step 2 with respect to x and y, respectively.

The second order partial derivative of f(x, y) with respect to x and x (f_xx) is obtained by differentiating the first order partial derivative with respect to x, obtained in Step 1, with respect to x. The second order partial derivative of -18x * cos^2 (3x ^ 2 - y) * sin (3x ^ 2 - y) with respect to x is -18 * cos^2 (3x ^ 2 - y) * sin (3x ^ 2 - y) - 36x * cos (3x ^ 2 - y) * sin (3x ^ 2 - y).

The second order partial derivative of f(x, y) with respect to y and y (f_yy) is obtained by differentiating the first order partial derivative with respect to y, obtained in Step 2, with respect to y. The second order partial derivative of -cos^3 (3x ^ 2 - y) with respect to y is 3cos^2 (3x ^ 2 - y) * sin (3x ^ 2 - y).

The second order partial derivative of f(x, y) with respect to x and y (f_xy) is obtained by differentiating the first order partial derivative with respect to x, obtained in Step 1, with respect to y. The second order partial derivative of -18x * cos^2 (3x ^ 2 - y) * sin (3x ^ 2 - y) with respect to y is 54x * cos^2 (3x ^ 2 - y) * cos (3x ^ 2 - y).

The second order partial derivative of f(x, y) with respect to y and x (f_yx) is obtained by differentiating the first order partial derivative with respect to y, obtained in Step 2, with respect to x. The second order partial derivative of -cos^3 (3x ^ 2 - y) with respect to x is 0, as the derivative of a constant with respect to x is always 0.

Therefore, the first order partial derivatives of f(x, y) are: ∂f/∂x = -18x * cos^2 (3x ^ 2 - y) * sin (3x ^ 2 - y) + (1 / (x * ln(y))) ∂f/∂y = -cos^3 (3x ^ 2 - y) + (-ln(x) / (y * ln(y) ^ 2))

And the second order partial derivatives of f(x, y) are: ∂^2f/∂x^2 = -18 * cos^2 (3x ^ 2 - y) * sin (3x ^ 2 - y) - 36x * cos (3x ^ 2 - y) * sin (3x ^ 2 - y) ∂^2f/∂y^2 = 3cos^2 (3x ^ 2 - y) * sin (3x ^ 2 - y) ∂^2f/∂x∂y = 54x * cos^2 (3x ^ 2 - y) * cos (3x ^ 2 - y) ∂^2f/∂y∂x = 0