Question

Find the first partial derivatives of the function f(x, y, z) = 4x sin(y - z).

Answer 1

The first partial derivatives of f(x, y, z) = 4x sin(y - z) with respect to x, y, and z are 4 sin(y - z), 4x cos(y - z), and -4x cos(y - z) respectively.

Step-by-step explanation:

The student asked to find the first partial derivatives of the function f(x, y, z) = 4x sin(y - z). To find the partial derivatives, we will differentiate the function with respect to each variable while treating the other variables as constants.

• The partial derivative with respect to x is df/dx = 4 sin(y - z) because sin(y - z) is a constant with respect to x.
• The partial derivative with respect to y is df/dy = 4x cos(y - z) because the derivative of sin with respect to y is cos(y - z), and we multiply by the coefficient 4x.
• The partial derivative with respect to z is df/dz = -4x cos(y - z) since the derivative of -sin with respect to z is -cos(y - z), and we multiply by the coefficient 4x.

Therefore, the first partial derivatives of the function f(x, y, z) are 4 sin(y - z), 4x cos(y - z), and -4x cos(y - z) respectively.