Question

How do you find the gradient of a function at a given point?

Answer 1

The gradient of a function $f(x,y,z)$ , in three dimensions is defined as the summation of the rate of change of $f$ in all these three directions separately keeping other two variables kept constant when calculating the rate of change for a particular.

Complete step-by-step answer:
The gradient of a function $f(x,y,z)$ , in three dimensions defined as :
$gradf(x,y,z)=\nabla f(x,y,z)=\dfrac{\partial f}{\partial x}i+\dfrac{\partial f}{\partial x}j+\dfrac{\partial f}{\partial x}k$
The gradient is a vector field of scalar function.
It is obtained by applying the vector operator $\nabla$ to the scalar function $f(x,y,z)$. Such a vector is called gradient or conservative field vector.
To interpret the gradient of a scalar field:
$gradf(x,y,z)=\nabla f(x,y,z)=\dfrac{\partial f}{\partial x}i+\dfrac{\partial f}{\partial x}j+\dfrac{\partial f}{\partial x}k$

Note: Its component in the $i$ direction is the partial derivative of $f$ with respect to $x$. This is the rate of change of $f$ in the $x$ direction since $y$ and $z$ are kept constant. In general, the component of $\nabla f$ in any direction is the rate of change of $f$ in that direction.