Whether the input space of f is two-dimensional, three-dimensional, or 1,000,000-dimensional: the gradient of f gives a vector in that input space that points in the direction that makes the …

The gradient is a way of packing together all the partial derivative information of a function. So let's just start by computing the partial derivatives of this guy.

The gradient of f, with our little del symbol, is a function of x and y. And it's a vector-valued function whose first coordinate is the partial derivative of f with respect to x.

Gradient is another word for slope. That means that finding the gradient (slope) involves a function and outputs a vector using nabla, which can be considered a function

Take a moment to delight in the fact that one single operation, the gradient, packs enough information to compute the rate of change of a function in every possible direction!

The way we compute the gradient seems unrelated to its interpretation as the direction of steepest ascent. Here you can see how the two relate.

When you expand it, the gradient would have five components, and the vector itself would have five components. So, this is the directional derivative and how you calculate it.

A more thorough look at the formula for directional derivatives, along with an explanation for why the gradient gives the slope of steepest ascent.

Key question: For a given vector-valued function v → (x, y) , how can we measure the change in density of particles around a point (x, y) as these particles flow along the vectors given by v → …

Multivariable calculus Course: Multivariable calculus > Unit 2 Lesson 2: Gradient and directional derivatives Gradient Finding gradients Gradient and graphs