Update Derivative.md

courses/Recipes/Common/Derivative/Derivative.md

@@ -14,16 +14,15 @@ Symbolic differentiation.
 
 #### Description
 
-Computing the [derivative](http://en.wikipedia.org/wiki/Differentiation_(mathematics)) of an expression with respect to some variable is a classical calculus problem. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity.
+Computing the [derivative](http://en.wikipedia.org/wiki/Differentiation_(mathematics)) of an expression with respect to some variable is a classical calculus problem. Loosely speaking, a derivative can be thought of as how much one quantity changes in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity.
 
-We present here rules for determining the derivative `dE/dX` of simple expressions `E` for a given variable `X`. Recall that for number `N`, variables `X` and `Y`, and expressions `E1` and `E2` the following rules apply:
-
-*  `dN / dX = 0`.
-*  `dX / dX = 1`.
-*  `dX / dY = 0`, when `X != Y`.
-*  `d(E1 + E2) /dX = dE1 / dX + d E2 /dX`.
-*  `d(E1 * E2) / dX =  (d E1 / dX  * E2) + (E1 * d E2 /dX)`.
+We present here rules for determining the derivative `dE/dX` of simple expressions `E` for a given variable `X`. Recall that for the number `N`, variables `X` and `Y`, and expressions `E1` and `E2` the following rules apply:
 
+* $\frac{dN}{dX} = 0$ 
+* $\frac{dX}{dX} = 1$
+* $\frac{dX}{dY} = 0$, when $X \neq Y$
+* $\frac{d(E1 + E2)}{dX} = \frac{dE1}{dX} + \frac{dE2}{dX}$
+* $\frac{d(E1 \cdot E2)}{dX} = \frac{dE1}{dX} \cdot E2 + E1 \cdot \frac{dE2}{dX}$
 
 #### Examples
 
@@ -65,20 +64,24 @@ test bool tstSimplity1() = simplify(mul(var("x"), add(con(3), con(5)))) == mul(v
 test bool tstSimplity2() = simplify(dd(E, var("x"))) == con(5);
 ```
 
+NOTE:
+* `con` stands for `constant` for example 1, 10, 99.
+* `var` stands for `variable` for example y, x, m.
+
 <1> Define a data type `Exp` to represent expressions.
 <2> Introduce an example expression `E` for later use.
 <3> Define the actual differentiation function `dd`. Observe that this definition depends on the use of patterns in function declarations, see [Rascal:Function].
 <4> Define simplification rules. 
 <5> A default rule is given for the case that no simplification applies.
-<6> Define the actual simplication function `simplify` that performs a bottom up traversal of the expression, applying simplification rules on ascend.
+<6> Define the actual simplification function `simplify` that performs a bottom-up traversal of the expression, applying simplification rules on ascending.
 
 
 Let's differentiate the example expression `E`:
 ```rascal-shell,continue
 dd(E, var("x"));
 ```
 As you can see, we managed to compute a derivative, but the result is far more complex than we would like.
-This is where simplification comes in. First try a simple case:
+This is where simplification comes in. First, try a simple case:
 ```rascal-shell,continue
 simplify(mul(var("x"), add(con(3), con(5))));
 ```