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Another view on this topic: think about the derivative of the MSE with respect to the inputs. You will need to apply the chain rule: $$\frac{dMSE}{dx_i}=\sum_i2(y_i-f(x_i))\cdot(-1)\cdot\frac{df(x_i)}{dx_i}$$ which only resembles the derivative of the parabola when, as mentioned in nbro's answer, $f(x_i)=c, \forall\ i$ or, more generally, when they $f(x)$ is ...