# Difference between gradient and derivative.

My question may be a bit stupid, but this morning I tried to explain the gradient to someone, and he makes a parallel with derivative of function $$f:\mathbb R\to \mathbb R$$. What he says is that for a function $$f:\mathbb R\to \mathbb R$$, the gradient and the derivative are the same. I agree that the scalar value are the same, but I’mnot sure that the meaning behind is the same.

For example, take $$f(x)=x^2$$. For me the gradient of $$f$$ is going to be the vector field $$\nabla f(x)=2x\cdot 1$$, where $$1$$ is the basis of $$\mathbb R$$, so it should look like that

whereas the derivative $$f'(x)$$ is really the rate of the function, and if it would be a vecteur field, it would be a vector field over the range of $$f$$, and not on the domain of $$f$$ as the gradient is. What do you think ?

To illustrate, I would say that the derivative field is in red and blue, and the gradient is in pink.