Find the first partial derivatives of the function.
f(x,y,z) = xe^(y/z)
a. fx = ?
b. fy = ?
c. fz = ?
x,y,z) = xe^(y/z)
a. fx = e^(y/z)
b. fy = xe^(y/z)/z
c. fz = -xye^(y/z)/z^2
To find the first partial derivatives of the function f(x,y,z) = xe^(y/z), we will differentiate the function with respect to each variable separately.
a. To find fx, we differentiate the function with respect to x while treating y and z as constants:
fx = (∂/∂x)(xe^(y/z))
Differentiating x with respect to x gives 1, and as y and z are constants, the derivative of e^(y/z) with respect to x is 0. Therefore,
fx = 1 * e^(y/z) = e^(y/z)
b. To find fy, we differentiate the function with respect to y while treating x and z as constants:
fy = (∂/∂y)(xe^(y/z))
Differentiating x with respect to y gives 0, and as x and z are constants, the derivative of e^(y/z) with respect to y is (1/z) * e^(y/z). Therefore,
fy = 0 * e^(y/z) + (1/z) * e^(y/z) = (1/z) * e^(y/z)
c. To find fz, we differentiate the function with respect to z while treating x and y as constants:
fz = (∂/∂z)(xe^(y/z))
Differentiating x with respect to z gives 0, and as x and y are constants, the derivative of e^(y/z) with respect to z is (-y/z^2) * e^(y/z). Therefore,
fz = 0 * e^(y/z) + (-y/z^2) * e^(y/z) = (-y/z^2) * e^(y/z)
Therefore, the first partial derivatives of the function f(x,y,z) = xe^(y/z) are:
a. fx = e^(y/z)
b. fy = (1/z) * e^(y/z)
c. fz = (-y/z^2) * e^(y/z)
To find the first partial derivatives of the function f(x, y, z) = xe^(y/z), we need to differentiate the function with respect to each variable separately while treating the other variables as constants.
a. To find fx, we differentiate the function f(x, y, z) with respect to x. Since x appears explicitly in the function, its derivative is simply 1 multiplied by the remaining terms.
So, fx = 1 * e^(y/z) = e^(y/z).
b. To find fy, we differentiate the function f(x, y, z) with respect to y. Using the chain rule, we need to differentiate the outer function (e^(y/z)) with respect to its inner function (y/z) and then multiply it by the derivative of the inner function with respect to y.
The derivative of e^(y/z) with respect to its inner function (y/z) is e^(y/z), and the derivative of y/z with respect to y is 1/z.
Therefore, fy = e^(y/z) * 1/z = e^(y/z) / z.
c. Similarly, to find fz, we differentiate the function f(x, y, z) with respect to z. Again, using the chain rule, we differentiate the outer function (e^(y/z)) with respect to its inner function (y/z) and then multiply it by the derivative of the inner function with respect to z.
The derivative of e^(y/z) with respect to its inner function (y/z) is e^(y/z), and the derivative of y/z with respect to z is -y/z^2.
Therefore, fz = e^(y/z) * (-y/z^2) = - y e^(y/z) / z^2.
So, the first partial derivatives of the function f(x, y, z) = xe^(y/z) are:
a. fx = e^(y/z)
b. fy = e^(y/z) / z
c. fz = - y e^(y/z) / z^2.