Find the total differential for
f(x,y,z)= cos(4x+3y+2z+1)
df = df/dx dx + df/dy dy + df/dz dz
let R = 4x+3y+2z+1
= -4sin(R)dx -3 sin(R) dy -2 sin(R)dz
= -sin(4x+3y+2z+1)(4 dx + 3 dy + 2 dz)
To find the total differential of the function f(x, y, z) = cos(4x + 3y + 2z + 1), we need to compute the partial derivatives with respect to each variable (x, y, z) and multiply them by the corresponding differentials (dx, dy, dz).
The partial derivative of f with respect to x (df/dx) is given by:
df/dx = -4sin(4x + 3y + 2z + 1)
The partial derivative of f with respect to y (df/dy) is given by:
df/dy = -3sin(4x + 3y + 2z + 1)
The partial derivative of f with respect to z (df/dz) is given by:
df/dz = -2sin(4x + 3y + 2z + 1)
Next, we multiply each partial derivative by the corresponding differential:
df = (df/dx) * dx + (df/dy) * dy + (df/dz) * dz
Therefore, the total differential of f(x, y, z) = cos(4x + 3y + 2z + 1) is:
df = -4sin(4x + 3y + 2z + 1) * dx - 3sin(4x + 3y + 2z + 1) * dy - 2sin(4x + 3y + 2z + 1) * dz
To find the total differential for the function f(x, y, z) = cos(4x + 3y + 2z + 1), we need to compute the partial derivatives of f with respect to each variable (x, y, and z), and then multiply them with the corresponding differentials dx, dy, and dz.
Let's start by finding the partial derivative of f with respect to x:
∂f/∂x = -sin(4x + 3y + 2z + 1) * ∂(4x + 3y + 2z + 1)/∂x
= -4sin(4x + 3y + 2z + 1)
Similarly, the partial derivative of f with respect to y is:
∂f/∂y = -sin(4x + 3y + 2z + 1) * ∂(4x + 3y + 2z + 1)/∂y
= -3sin(4x + 3y + 2z + 1)
And the partial derivative of f with respect to z is:
∂f/∂z = -sin(4x + 3y + 2z + 1) * ∂(4x + 3y + 2z + 1)/∂z
= -2sin(4x + 3y + 2z + 1)
Now, to find the total differential df, we multiply each partial derivative by the corresponding differential:
df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz
= -4sin(4x + 3y + 2z + 1)dx - 3sin(4x + 3y + 2z + 1)dy - 2sin(4x + 3y + 2z + 1)dz
So, the total differential for f(x, y, z) = cos(4x + 3y + 2z + 1) is -4sin(4x + 3y + 2z + 1)dx - 3sin(4x + 3y + 2z + 1)dy - 2sin(4x + 3y + 2z + 1)dz.