what is a gradient
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Gradient is defined as vertical rise/ horizontal run.
A gradient is a mathematical concept that measures the rate of change of a function in multiple dimensions. It represents the direction and magnitude of the steepest ascent or descent of a function at a given point.
To compute the gradient of a function, you need to calculate the partial derivative of the function with respect to each of its input variables. The partial derivative of a function measures how much the function changes when only one of its variables changes, keeping the other variables constant.
Here is the step-by-step process to calculate the gradient of a function:
1. Identify the function for which you want to find the gradient. For example, let's say we have a function f(x, y) = x^2 + 2y.
2. Calculate the partial derivative of the function with respect to each variable. For f(x, y), we calculate the partial derivatives ∂f/∂x and ∂f/∂y separately.
The partial derivative ∂f/∂x measures how much f(x, y) changes when x changes, while keeping y constant. In this example, ∂f/∂x = 2x.
The partial derivative ∂f/∂y measures how much f(x, y) changes when y changes, while keeping x constant. In this example, ∂f/∂y = 2.
3. Combine the partial derivatives into a vector. In this example, the gradient vector, denoted as ∇f, would be (∂f/∂x, ∂f/∂y) = (2x, 2).
The gradient is a useful concept in many areas of mathematics and is especially important in optimization algorithms, where it is used to find the minimum or maximum values of functions.