Differentiation Formulas

Serial	Expression	Derivative	Remarks
	u+v	du/dx+dv/dx
	u-v	du/dx-dv/dx
	u/v	(v*du/dx-u*dv/dx)/v^2
	u*v	u*dv/dx+v*du/dx
	c*u	c*du/dx	c is constant
	u^v	v*u^(v-1)*du/dx+u^v*ln(u)*dv/dx
	u^n	n*u^(n-1)*du/dx	n is real
	c^u	c^u*ln(c)*du/dx	c is constant
	e^u	e^u*du/dx	e = 2.7182818284590452353602874713527
1	sin(u)	cos(u)*du/dx
2	cos(u)	-sin(u)*du/dx
3	tan(u)	sec(u)^2*du/dx
4	sec(u)	sec(u)*tan(u)*du/dx
5	cosec(u)	-cosec(u)*cot(u)*du/dx
6	cot(u)	-cosec(u)^2*du/dx
7	sinh(u)	cosh(u)*du/dx
8	cosh(u)	sinh(u)*du/dx
9	tanh(u)	sech(u)^2*du/dx
10	sech(u)	sech(u)*tanh(u)*du/dx
11	cosech(u)	cosech(u)*coth(u)*du/dx
12	coth(u)	-cosech(u)^2*du/dx
13	asin(u)	1/sqrt(1-u^2)*du/dx
14	acos(u)	-1/sqrt(1-u^2)*du/dx
15	atan(u)	1/(1+u^2)*du/dx
16	asec(u)	1/(	u
17	acosec(u)	-1/(	u
18	acot(u)	-1/(1+u^2)*du/dx
19	asinh(u)	1/sqrt(u^2+1)*du/dx
20	acosh(u)	1/sqrt(u^2-1)*du/dx
21	atanh(u)	1/(1-u^2)*du/dx
22	asech(u)	-1/(u*sqrt(1-u^2))*du/dx
23	acosech(u)	-1/(u*sqrt(1+u^2))*du/dx
24	acoth(u)	1/(1-u^2)*du/dx
25	sqrt(u)	1/(2*sqrt(u))*du/dx
26	log10(u)	1/(u*ln(10))*du/dx
27	log(u)	1/u*du/dx
28	ln(u)	1/u*du/dx
29	sign(u)	0
30	abs(u)	u/	u

Mathematical Expression

A mathematical expression includes numbers, operators, functions, variables and operators. Operators include +, -, *, / and ^. Functions are like sin(x) and variables are like x. Our target here is to identify each element and to get the right differentiation rule applied to that element or a group of elements.

Expression examples:

sin(2*x)/x

sin(45+cos(2)/tan(x)

sign(cos(x)

32*9-8/2

Differentiation Steps{#steps}

Expression Parsing to Fill Stack

Expression parsing is the process of scanning an expression to find its elements (numbers, operators, functions and variables). Each operator should have two operands; for example, take 2*12. The operator "*" has two operands: 2 and 12. Each function has its argument. For example, for sin(x), the function is sin and the argument is x. So, we should scan the expression in a specific way to identify each operator and the operands, as well as each function argument, to simplify the calculation or the differentiation.

Expression execution depends on operator precedence. If we can construct a stack of operators and operands such that each operator is followed by its two operands, 2*12 should construct this stack: "*, "2", "12". So, the calculation function checks the stack. If the item is an operator, then it takes its two operands and does the calculation for them in a recursion formula, so the whole expression can be calculated. For example, the expression:

sin(2\*12)/7+9^2

...should be calculated in these steps:

1. Calculate sin(2*12)/7

2. Calculate 9^2

3. Then apply the operator + for the two results

4. To calculate point 1, we need to:

   1. Calculate sin(2*12)

   2. Calculate 7

   3. Then divide the two results

   4. To calculate point 4.1, we need to:

      1. Calculate 2*12

      2. Apply the sin function to the result

      3. To calculate point 4.4.1, we need to:

         1. Apply the operator * to the two operands 2 and 12

It is obvious now that it is a recursion operation. To do that, we need to formulate the expression in a stack of operators and operands, and process that stack in a certain way to get the final result. The main steps to do that in brief are:

• Find operators

• Split the expression at the operator index

• Add the two resultant operands to the stack

• If operators are not found, then check if the expression starts with a function name

If we apply the previous steps, we will have the stack in this form:

+
/
sin(2*12)
7
^
9
2

The mathematical expression parsing is as follows:

• Insert the first input into the stack

• Loop in the expression stack to check if any expression can be divided into two queries

• Check if the expression item is an operator (+-*/^)

• Parse the expression to find the operators

• If an operator is found:

  • Split the expression into two queries at the operator index

  • Add the left operand of the operator as a new expression

  • Add the right operand of the operator as a new expression

• Else

  • Check if the expression string starts with a function or a parenthesis

  • If a function is found, set its number in the expression data

Apply Differentiation Formulas on the Stack

Applying differentiation formulas means to iterate the stack, take each operator and apply the differentiation rule - depending on the operator

• on the two operands of the operator:

Expression	Derivative
u+v	du/dx+dv/dx
u-v	du/dx-dv/dx
u/v	(v*du/dx-u*dv/dx)/v^2
u*v	u*dv/dx+v*du/dx
c*u	c*du/dx
u^n	n*u^(n-1)*du/dx
c^u	c^u*ln(c)*du/dx
e^u	e^u*du/dx

If any operand is an operator, then the function is called again in a recursive way to differentiate the whole expression. The following code represents this function:

Optimize Equation

Optimizing the equation includes these simple steps:

• Replace "--" with "" or "+"

• Replace "+-" with "-"

• Replace "((....))" with "(....)"

• Remove any "1*"

• Remove any "*1"

• Remove any "^1"

• Remove any "1*"

• Remove unneeded parentheses

Differentiation Pseudo Code

Differentiate(input)

{

Stack = FillStack(input)

output = DifferentiateStack(Stack)

Optimize(output)

return output

}

FillStack(input)

{

operators\[\] { "+-", "\*/", "^%" }

stack.push(input)

loop( n = 1 to stack.size() )

{

if stack\[n\] is not operator

if GetOperator(stack\[n\], operators) success

{

Split stack\[n\]

stack.Insrt(left operand)

stack.Insrt(right operand)

}

else

GetFunction(stack\[n\])

}

}

DifferentiateStack(stack, index)

{

if stack\[index\] is operator

{

index++

u = stack\[index\]

du = DifferentiateStack(stack, index)

v = stack\[index\]

dv = DifferentiateStack(stack, index)

if operator = '-' or '+'

output = du+operator+dv

else if operator = '\*'

output = u\*dv+du\*v

else if operator = '/'

output = (du\*v-u\*dv)/v^2

else if operator = '^'

output = v\*u^(v-1)\*du+u^v\*ln(u)\*dv

}

else

output = stack\[index++\].GetDifferentiation()

return output

}

void Optimize(str)

{

replace "--" with "" or "+"

replace "+-" with "-"

replace "((....))" with "(....)"

remove any 1\*

remove any \*1

remove any exponent equal 1

remove unneeded parentheses

if str changed then

Optimize(str)

}

ExpressionItem::GetDifferentiation()

{

if Function then

{

arument = Argument(input);

if function = SIN

output = Differentiate(arument)\*cos(arument)

else if function = COS

output = Differentiate(arument)\*(-sin(arument))

else

...

}

}

else

{

if input = "x"

output = "1"

else if input = "-x"

output = "-1"

else if input is numeric

output = "0"

else

output = "d"+input+"/dx"

}

}

The Differentiation tab displays the differentiation results after applying the differentiation steps in the input expression. The input expression is calculated before differentiation to optimize any arithmetic part that includes numbers as in the example:

e^sin(pi/3)/tan(x) is optimized to 2.377442/tan(x) before differentiation to get the result (-2.377442*sec(x)^2)/(tan(x))^2.

Differentiation Stack

Calculation

This tab is a Symbolic Calculator that can calculate any mathematical expression, however complicated. The calculation includes only two constants:

e = 2.7182818284590452353602874713527

pi = 3.1415926535897932384626433832795

Calculation Stack

This view is used to view the stack of the calculation details. This means "the calculation of." The stack contents are arranged from top to bottom. Calculation steps are differentiation steps, except the operators' formulas. The advantage of this view is that it enables showing of the parts that can be calculated in the expression and those that can't be calculated, as in the example in the figure.

Draw

This tab is used to view the input function curve and its differentiation curve. You can view the input function curve only by clicking "Draw." The view includes horizontal and vertical axes, as well as grid lines. You move over any point in the view to see its coordinates in the left upper corner, and you can zoom in on any part by selecting it with the mouse. The horizontal and vertical axes' limits update automatically to include the input function points in the view. You can change these limits by right clicking the view.

Points of Interest

1. Part of Expression Calculation

The advantage of my expression parsing over other parsers on the net is that it can calculate part of the expression and leave the other part, and that includes any variables. For example, take the expression sin(45+sin(2))/tan(x). If you try to input this expression to any free parser on the net, it will give an error indicating an invalid character x. However, my parser gives you 0.937227/tan(x). For any complex expression, my parser tries to optimize the expression before the differentiation, and calculates any part that can be calculated. So, the differentiation of sin(45+sin(2))/tan(x) is (-0.937227*sec(x)^2)/(tan(x))^2.

1. Operator Precedence{#precedence}

When a complex expression has multiple operators, the operator precedence determines the sequence in which the operations are performed. The order of execution can significantly affect the resulting value. Operators have these precedence levels. An operator on higher levels is evaluated before an operator on a lower level:

• "+" (Positive), - (Negative)

• ^ (Exponent), % (Modulo)

• * (Multiply), / (Division)

• • (Add), - (Subtract)

When two operators in an expression have the same operator precedence level, they are evaluated left to right based on their position in the expression. For example, in 4 - 2 + 27, the subtraction operator is evaluated before the addition operator to get 2 + 27, which yields an expression result of 29.

Use parentheses to override the defined precedence of the operators in an expression. Everything within the parentheses is evaluated first to yield a single value before that value can be used by any operator outside of the parentheses. For example, 2 * 4 + 5 evaluates to 8 + 5, which yields an expression result of 13. The expression 2 * (4 + 5) evaluates to 2 * 9; the parentheses cause the addition to be performed first and so the expression result is 18.

If an expression has nested parentheses, the most deeply nested expression is evaluated first. The example 2 * (4 + (5 - 3) ) contains nested parentheses, with the expression 5 - 3 in the most deeply nested set of parentheses. This expression yields a value of 2. Then the addition operator (+) adds this result to 4, which yields a value of 6. Finally, the 6 is multiplied by 2 to yield an expression result of 12.

1. Zoom In

In the drawing area, you can select any area to zoom in on. For example, the function sin(100*x) has a very large variation, as in the following figure:

By zooming in many times, we can view the smaller details, as in the following figures:

Note: Check the horizontal axis resolution after each zoom-in.