Gradient of a Function at a Point

Find the gradient of the function $f(x, y) = 3x^2 + y + 6$ at the point $(1, -1)$.

Calculating the gradient of a function is a fundamental concept in multivariable calculus. It provides significant insight into the behavior of a function across different points in its domain. When you are asked to find the gradient of a function such as $f(x, y) = 3x^2 + y + 6$, you are essentially determining the vector that points in the direction of the greatest rate of increase of the function. The vector's components are partial derivatives with respect to each variable of the function. This is a simple yet powerful tool often used in optimization problems or when analyzing how a function behaves in 3D space.

In this particular problem, you are asked to find the gradient at a specific point, $(1, -1)$. This involves computing the partial derivatives of the function with respect to each variable and then evaluating these derivatives at the given point. Understanding this process is crucial for extending the concept of gradient beyond two-variable functions to more complex functions involving multiple variables. Additionally, the gradient vector plays a key role in defining tangent planes and approximating functions locally by linearization, which are pivotal in advanced topics in multivariable calculus and differential equations. Understanding and visualizing these concepts help to deepen the grasp of calculus, especially when dealing with real-world applications where functions of several variables are common.

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