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Last updated on October 6, 2025
In mathematics, the gradient is a vector that describes the direction and rate of the steepest increase of a function. It is a vital concept in calculus and differential geometry. In this topic, we will learn the formula for the gradient.
The gradient measures the rate and direction of change in a function. Let’s learn the formula to calculate the gradient.
The gradient of a function, often denoted as ∇f, is a vector of its partial derivatives.
For a function f(x, y, z), the gradient is calculated as: ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)
This represents the rate and direction of change in the function.
In math and real life, the gradient formula is used to analyze and understand changes in multivariable functions. Here are some important uses of the gradient:
Students might find the gradient formula tricky and confusing. Here are some tips and tricks to master the gradient formula:
In real life, the gradient plays a major role in understanding changes in multivariable functions. Here are some applications of the gradient formula:
Students make errors when calculating gradients. Here are some mistakes and the ways to avoid them, to master the gradient formula.
Find the gradient of f(x, y) = x² + y²?
The gradient is (2x, 2y)
For f(x, y) = x² + y², the partial derivative with respect to x is 2x, and with respect to y is 2y. Thus, the gradient ∇f = (2x, 2y).
Find the gradient of f(x, y, z) = xyz?
The gradient is (yz, xz, xy)
For f(x, y, z) = xyz, the partial derivative with respect to x is yz, with respect to y is xz, and with respect to z is xy. Thus, the gradient ∇f = (yz, xz, xy).
Calculate the gradient of f(x, y) = 3x^2y + 4y^3?
The gradient is (6xy, 3x² + 12y²)
For f(x, y) = 3x²y + 4y², the partial derivative with respect to x is 6xy, and with respect to y is 3x² + 12y². Thus, the gradient ∇f = (6xy, 3x² + 12y²).
What is the gradient of f(x, z) = 5xz² + 7x?
The gradient is (z² + 7, 10xz)
For f(x, z) = 5xz² + 7x, the partial derivative with respect to x is z² + 7, and with respect to z is 10xz. Thus, the gradient ∇f = (z² + 7, 10xz).
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.