Cost function in backpropogation section is confusing and possibly incorrect. about ml-glossary

[sprakshith]

The factor of (1/2) is often included for mathematical convenience. When you take the derivative of the cost function to perform gradient descent (a common optimization algorithm), the (1/2) cancels out when computing the gradient, simplifying the expressions and computations.

Also I believe that this cost function is just for an instance and they have not yet taken the mean in this case:

Equation for Mean Squared Error with (1/2) multiplied for convenience: $$\frac{1}{2n} \sum_{i=1}^{n} \left( \hat{y}^{(i)} - y^{(i)} \right)^2$$

And then after taking the differentiation: $$\frac{1}{n} \sum_{i=1}^{n} \left( \hat{y}^{(i)} - y^{(i)} \right) \cdot x^{(i)}$$

PS: If I am wrong please correct me, thanks.