Find all of the first partial derivatives of f(x,y,z)=arctan y/xz
Remember - treat the other variables like constants.
Since d/dx arctan(u) = u'/(1+u^2)
Fx = (-y/zx^2)/(1+ y^2/x^2z^2)
Fy = (1/xz)//(1+ y^2/x^2z^2)
Fz = -y/xz^2)/(1+ y^2/x^2z^2)
you can massage them a bit if you like.
To find the first partial derivatives of the function f(x,y,z) = arctan(y/xz), we will differentiate with respect to each variable one at a time while treating the other variables as constants.
1. Partial derivative with respect to x:
To find the partial derivative of f(x,y,z) with respect to x, we differentiate f(x,y,z) with respect to x while treating y and z as constants.
∂f/∂x = ∂(arctan(y/xz))/∂x
Using the chain rule, we have
∂f/∂x = (∂(arctan(y/xz))/∂(y/xz)) * (∂(y/xz)/∂x)
= (1/(1+(y/xz)^2)) * (∂(y/xz)/∂x)
Differentiating y/xz with respect to x, we get
∂(y/xz)/∂x = -yz/x^2z
Therefore, the first partial derivative with respect to x is
∂f/∂x = (1/(1+(y/xz)^2)) * (-yz/x^2z)
= -yz/(x^2z + y^2)
2. Partial derivative with respect to y:
To find the partial derivative of f(x,y,z) with respect to y, we differentiate f(x,y,z) with respect to y while treating x and z as constants.
∂f/∂y = ∂(arctan(y/xz))/∂y
Using the chain rule, we have
∂f/∂y = (∂(arctan(y/xz))/∂(y/xz)) * (∂(y/xz)/∂y)
= (1/(1+(y/xz)^2)) * (∂(y/xz)/∂y)
Differentiating y/xz with respect to y, we get
∂(y/xz)/∂y = 1/xz
Therefore, the first partial derivative with respect to y is
∂f/∂y = (1/(1+(y/xz)^2)) * (1/xz)
= 1/(xz(1+(y/xz)^2))
3. Partial derivative with respect to z:
To find the partial derivative of f(x,y,z) with respect to z, we differentiate f(x,y,z) with respect to z while treating x and y as constants.
∂f/∂z = ∂(arctan(y/xz))/∂z
Using the chain rule, we have
∂f/∂z = (∂(arctan(y/xz))/∂(y/xz)) * (∂(y/xz)/∂z)
= (1/(1+(y/xz)^2)) * (∂(y/xz)/∂z)
Differentiating y/xz with respect to z, we get
∂(y/xz)/∂z = -y/xz^2
Therefore, the first partial derivative with respect to z is
∂f/∂z = (1/(1+(y/xz)^2)) * (-y/xz^2)
= -y/(xz(1+(y/xz)^2))
Hence, the first partial derivatives are:
∂f/∂x = -yz/(x^2z + y^2)
∂f/∂y = 1/(xz(1+(y/xz)^2))
∂f/∂z = -y/(xz(1+(y/xz)^2))
To find the first partial derivatives of a function, we differentiate the function with respect to each variable while treating the other variables as constants.
In this case, we have the function f(x, y, z) = arctan(y/(xz)).
To find the partial derivative with respect to x, we differentiate the function with respect to x while treating y and z as constants.
Partial derivative with respect to x:
∂f/∂x = d/dx[arctan(y/(xz))]
To differentiate arctan(y/(xz)), we can use the chain rule. Let u = y/(xz):
∂f/∂x = d/dx[arctan(u)]
= (1/(1+u^2)) * du/dx
To find du/dx, we differentiate u with respect to x:
du/dx = d/dx[y/(xz)]
To differentiate y/(xz), we use the quotient rule:
du/dx = [(d/dx[y])(xz) - y(d/dx[xz])]/(xz)^2
Differentiating y and xz with respect to x:
du/dx = [(0)(xz) - y((d/dx[x])(z) + x(d/dx[z]))]/(xz)^2
= [- yz - x(d/dx[z])]/(xz)^2
Since d/dx[z] = 0 (since z is a constant with respect to x), du/dx simplifies to:
du/dx = - yz/(xz)^2
Now we substitute this value back into the expression for ∂f/∂x:
∂f/∂x = (1/(1+u^2)) * (- yz/(xz)^2)
= - yz/(xz)^2 * 1/(1+(y/(xz))^2)
= - yz/(xz)^2 * 1/(1+y^2/(x^2z^2))
= - yz/(xz)^2 * x^2z^2/(x^2z^2+y^2)
= - yz/(x^2z^2+y^2)
Therefore, the partial derivative of f(x, y, z) with respect to x is - yz/(x^2z^2+y^2).
Similarly, we can find the partial derivatives with respect to y and z.
Partial derivative with respect to y:
∂f/∂y = d/dy[arctan(y/(xz))]
= (1/(1+u^2)) * du/dy
To find du/dy, we differentiate u with respect to y:
du/dy = d/dy[y/(xz)]
Differentiating y with respect to y:
du/dy = [(d/dy[y])(xz) - y(d/dy[xz])]/(xz)^2
= [xz - x(d/dy[z])]/(xz)^2
= [xz]/(xz)^2
= 1/(xz)
Substituting this value back into the expression for ∂f/∂y:
∂f/∂y = (1/(1+u^2)) * (1/(xz))
= 1/(xz(1+(y/(xz))^2))
= xz/(xz^2+y^2)
Therefore, the partial derivative of f(x, y, z) with respect to y is xz/(xz^2+y^2).
Partial derivative with respect to z:
∂f/∂z = d/dz[arctan(y/(xz))]
= (1/(1+u^2)) * du/dz
To find du/dz, we differentiate u with respect to z:
du/dz = d/dz[y/(xz)]
Differentiating xz with respect to z:
du/dz = [(d/dz[y])(xz) - y(d/dz[xz])]/(xz)^2
= [(d/dz[y])(xz) - y(d/dz[x])(z) - yx]/(xz)^2
= [0 - y - yx]/(xz)^2
= - (y + yx)/(xz)^2
= - y(1 + x)/(xz)^2
Substituting this value back into the expression for ∂f/∂z:
∂f/∂z = (1/(1+u^2)) * (- y(1 + x)/(xz)^2)
= - y(1 + x)/(xz)^2 * 1/(1+(y/(xz))^2)
= - y(1 + x)/(xz)^2 * x^2z^2/(x^2z^2+y^2)
= - y(1 + x)/(x^2z^2+y^2)
Therefore, the partial derivative of f(x, y, z) with respect to z is - y(1 + x)/(x^2z^2+y^2).
The first partial derivatives of f(x, y, z) are:
∂f/∂x = - yz/(x^2z^2+y^2)
∂f/∂y = xz/(xz^2+y^2)
∂f/∂z = - y(1 + x)/(x^2z^2+y^2)