Question

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

Answer 1

Hint: 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.