Proving the laws of deduction

### Introduction Although we have previously argued that proving the laws of derivation is quite unnecessary for economists, we nevertheless found it appropriate to illustrate the way of mathematical thinking in the context of the laws of derivation. As an example, we will refer to the function `f(x)=x^2`. ### Calculating the slope of the function at some x value We would like to calculate the slope at point B. The point is `(X_B , X^2_B)`. Point A is close to B and its values ​​are `(X_A , X^2_A)`. The explanation is accompanied by the diagram below. ### Symbols – The value `X_A` is the value that is tangent to `X_B` on the right or left.
– Point A is the result of the function at the value `X_A`, that is, `X^2_A`.
– Point B is the result of the function at the value `X_B`, i.e. `X^2_B`.
– `Deltay` is defined as `[f(X_A),f(X_B)] `, meaning `[X^2_A,X^2_B] `.
– `Deltax` is defined as `[X_A-X_B] `. The fraction `(Deltay)/(Deltax)` when `X_A=>X_B` represents the slope of the function at point B (in words: when `X_A` tends to `X_B` and point A converges to point B). We substitute `Deltay`, the expression `[X^2_A-X^2_B] ` and instead of `Deltax` the expression `[X_A-X_B] ` and we get **Equation 1**. **Equation 1**: `[(X^2_A-X^2_B)/(X_A-X_B)]=(Deltay)/(Deltax)` We will refer to the term in brackets in Equation 1: When `X_A=>X_B` `X_A` converges with `X_B` and the point A converges with B. If, following the convergence, we substitute `X_B` in place of `X_A`, we will get 0 in the numerator and 0 in the denominator, and the result of the fraction is 0 `[0/0=] `, which means: the slope at every point on the curve is 0. We of course know that this is not true. ### Sophistication Mathematicians break the numerator into 2 products: `(X_A-X_B)(X_A+X_B)=[X^2_A-X^2_B] ` and we obtain **Equation 2**. **Equation 2**: `((X_A-X_B)(X_A+X_B))/(X_A-X_B)=(Deltay)/(Deltax)` If we now substitute `X_B` in place of `X_A` we obtain `2X_B`. It therefore follows that when `f(x)=x^2` the derivative is: `f'(x)=2x` Similar intricacies are used in proving the other derivative laws.