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 enter image description here

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.

enter image description here