Gradient of a function

Question

Gradient of a function is $\langle f_x(x,y),f_y(x,y) \rangle$. But I don't understand this gradient vector shows what. When I find gradient of some function, that vectors represents what? Thank you.

Answer 1

The gradient is a fancy word for derivative, or the rate of change of a function. It’s a vector (a direction to move) that

1. Points in the direction of greatest increase of a function.
2. Is zero at a local maximum or local minimum (because there is no single direction of increase).

The term “gradient” is typically used for functions with several inputs and a single output (a scalar field).

We know the definition of the gradient: a derivative for each variable of a function.

Suppose,

If we want to find the direction to move to increase our function the fastest, we plug in our current coordinates (such as 3,4,5) into the equation and get:

So, this new vector (1, 8, 75) would be the direction we’d move in to increase the value of our function.

The other use of gradient is to find the max/min of multivariable functions.

Answer 2

For a multivariable function $f(x,y)$, that vector is called $\textbf{grad}f$, the gradient of $f$, sometimes denoted $\nabla f$.

It's a vector that points in the direction of greatest increase of the function.