Differentiation of a variable with respect to itself

Let $y = x$.

Following the General Rule, §4.7, we have

• FIRST STEP. $y + \Delta y = x + \Delta x$.

• SECOND STEP. $\Delta y = \Delta x$

• THIRD STEP. $\frac{\Delta y}{\Delta x} = 1$.
• FOURTH STEP. $\frac{dy}{dx} = 1$.

Therefore, $\frac{dy}{dx} = 1$ (equation (II) above). The derivative of a variable with respect to itself is unity.