Find the gradients of the following functions:(a) f (x, y, z) = x2 + y3 + z4 .(b) f (x, y, z) = x2 y3 z4 .(c) f (x, y, z) = e x sin(y) ln(z)
Question
Find the gradients of the following functions:(a) f (x, y, z) = x2 + y3 + z4 .(b) f (x, y, z) = x2 y3 z4 .(c) f (x, y, z) = e x sin(y) ln(z)
Solution
Sure, let's find the gradients of the given functions.
(a) f (x, y, z) = x^2 + y^3 + z^4
The gradient of a function is a vector of its partial derivatives. So, we need to find the partial derivatives of f with respect to x, y, and z.
∂f/∂x = 2x ∂f/∂y = 3y^2 ∂f/∂z = 4z^3
So, the gradient of f is (2x, 3y^2, 4z^3).
(b) f (x, y, z) = x^2 y^3 z^4
Again, we find the partial derivatives:
∂f/∂x = 2xy^3z^4 ∂f/∂y = 3x^2y^2z^4 ∂f/∂z = 4x^2y^3z^3
So, the gradient of f is (2xy^3z^4, 3x^2y^2z^4, 4x^2y^3z^3).
(c) f (x, y, z) = e^x sin(y) ln(z)
The partial derivatives are:
∂f/∂x = e^x*sin(y)ln(z) ∂f/∂y = e^xcos(y)ln(z) ∂f/∂z = e^xsin(y)/z
So, the gradient of f is (e^x*sin(y)ln(z), e^xcos(y)ln(z), e^xsin(y)/z).
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