Find $\nabla f$ at the given point.
$f (x, y, z) = x^{3} + y^{3} - 4 z^{2} + z \ln x$
${\nabla f |}_{(1, 2, 3)} = ◻ i + (◻) j + (◻) k (Simplify your answers.)$

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Answer

$\nabla f |_{(1, 2, 3)} = 6 i + 12 j - 24 k$

Steps

Step 1 ：Find the partial derivatives of the function: $f (x, y, z) = x^{3} + y^{3} - 4 z^{2} + z \ln x$

Step 2 ： $\frac{\partial f}{\partial x} = 3 x^{2} + \frac{z}{x}$

Step 3 ： $\frac{\partial f}{\partial y} = 3 y^{2}$

Step 4 ： $\frac{\partial f}{\partial z} = - 8 z + \ln x$

Step 5 ：Evaluate the partial derivatives at the point (1, 2, 3):

Step 6 ： ${\frac{\partial f}{\partial x} |}_{(1, 2, 3)} = 3 (1)^{2} + \frac{3}{1} = 6$

Step 7 ： ${\frac{\partial f}{\partial y} |}_{(1, 2, 3)} = 3 (2)^{2} = 12$

Step 8 ： ${\frac{\partial f}{\partial z} |}_{(1, 2, 3)} = - 8 (3) + \ln 1 = - 24$

Step 9 ： $\nabla f |_{(1, 2, 3)} = 6 i + 12 j - 24 k$