# Gradient of an interpolated function

Can anyone please give a explanation: what do you mean by gradient of an interpolated function?

Suppose, $$f(x, y, z) = 2x^3 + 3y^2 -z$$ is a function, and one result of the interpolation for the position $$(0.9, 0.4, 0.6)$$ is $$4$$. Actually, I want to know, does the concept of normal gradient and gradient of interpolated function is the same for a function?

Suppose for a function, normal gradient in the $$x$$ direction is $$\frac{\partial f}{\partial x}\bigg|_{x = x_1} = \lim_{x_2 \to x_1} \frac{f(x_2, y, z) - f(x_1, y, z)}{x_2 - x_1}$$

An interpolated function is a function through a given set of points. An example of such a point would be: $$f(0.9,0.4,0.6) = 4$$. But apart from this property the interpolated function should be treated as any other function. This implies that the gradient of such a function is still the same as for any other.
$$\nabla f(x,y,z)=\big(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}\big)$$
In your case: $$∇f(x,y,z)=\big(6x^2,6y,-1\big)$$