Find the gradient of f (x, y) = x2 − 3(y − 1)2 + 3.∇ f = 2xi − 6(y − 1)j.
Question
Find the gradient of f (x, y) = x2 − 3(y − 1)2 + 3.∇ f = 2xi − 6(y − 1)j.
Solution
To find the gradient of the function f(x, y) = x^2 - 3(y - 1)^2 + 3, we need to take the partial derivatives with respect to x and y.
The partial derivative of f with respect to x is found by treating y as a constant and differentiating with respect to x.
∂f/∂x = 2x
The partial derivative of f with respect to y is found by treating x as a constant and differentiating with respect to y.
∂f/∂y = -6(y - 1)
Therefore, the gradient of f, denoted ∇f, is given by the vector (2x, -6(y - 1)).
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