Is the gradient of a function basically it's derivative?
For instance, given f(x) = 3x^4 + 6x^2 + 7x + 3, the gradient of f(x) would be f'(x) = 12x^3 + 12x + 7, right?
The gradient of a (scalar) field is a vector field. Suppose you have a scalar field, such as a temperature field $\displaystyle \phi=\phi(x,y,z)$. Then you can form a vector field $\displaystyle (\nabla \phi)(x,y,z)=\left( \frac{\partial \phi}{\partial x},\frac{\partial \phi}{\partial y},\frac{\partial \phi}{\partial z}\right)$ called the gradient of phi.
So it is not the derivative but it is deeply related