Question

Compute the partial derivative: f(x,y)=sin(x3−6y)fy​(0,π)=​

Answer 1

The partial derivative of the function − 6 · cos(x^3 − 6y) with respect to y at the point (0, π) is − 6.

Step-by-step explanation:

To compute the partial derivative of f(x,y) = sin(x3 − 6y) with respect to y, denoted as f_y, we take the derivative of the function f with respect to y, while treating x as a constant:

f_y(x, y) = ∂f/∂y = − 6 · cos(x3 − 6y)

To find f_y(0, π), we substitute x = 0 and y = π into our expression for f_y:

f_y(0, π) = − 6 · cos(0 − 6π) = − 6 · cos(-6π)

Since the cosine function is periodic with period 2π, we know that cos(-6π) = cos(0) = 1. Thus, f_y(0, π) = − 6 · 1 = − 6.

Therefore, the value of the partial derivative f_y at the point (0, π) is − 6.