Derivatives o Transcendal Functions

Using previously published formulas:

Don't be confused when you see trigonometric functions, it's very easy, using each formula given, you just have to identify each function and follow the steps as shown below:

Arrange as first the derivative of the argument, then the trigonometric function that is specified in formulas, and  put the argument that is not touched.
As you can see you can even combine normal  function with trigonometric functions. 

Here are other examples to understand better.

Easy :D

This topic can continue using the product rule. Exactly the same steps.

It can be this way or simplify terms.

It also continues to using the quotient rule.

Natural logarithms:

They are very easy, just follow the formula previously given:

          It is the derivative of the argument as the numerator and the denominator is the argument.

Derivatives by the Four Steps Procedure

"Derivative"

This procedure consists of four steps that will always be the same.

1.- Add the increments in (X)

2.- Subtract the original function.

3.- Divide by the increment

4.- Limit when the increment is 0

En este caso, se puede sustituir con 0

Increment is always the same

Last Example

The Chain Rule

The chain rule is a way to derive, mostly used in binomials or trinomials with some exponent.

"Derivative of the outside evaluated at the inside times the derivative of the inside"

Examples

In simple words, you put the exponent outside, you  never touch the argument, you put it exactly. Then you do the derivative of the argument.

Make a little algebra and it's done.

Don't forget to subtract 1 to the exponent.

Here you also have the option to change the roots.

To get the derivatives of these types of functions, you have to use product or quotient rules.

Basic Derivatives

* Let's use some formulas previously published. *

In mathematics, the derivative of a function is a measure of the speed with which changes the value of the mathematical function, as you change the value of the independent variable.

The derivative is the slope of the tangent line to the graph at point "X"

The easiest derivatives

If is a single number, no matter what, it always be 0

If is a number comes with a variable,  it always be the simple number without the variable.

If is a number comes with a variable and a exponent, you need to multiply the exponent times the number, and subtract 1 to the exponent.

It is exactly the same, only several times.

In this example, in order to don't get confused with the root, you can do what is shown.

In this you have to use the product rule. do the correct algebra.

In this you have to use the quotient rule (divisions), make the correct algebra.

Limits

A limit is a fixed magnitude which approximates more and more to the terms of an infinite sequence.

"The relation (F) of the elements of a set (A) with the elements of one set (B). Proximity between a value and a point"

Examples, Solve the exercise

As you can see, is a substitution. We have a limit, but we want to know  the value of the

 function (x) when it approaches -1 or 1/2

There are also cases where we need to factorize 

And then we can divide.

Next example:

This example is a bit more complicated, because we have root, but if we multiply ALL the function by  the root, we can cancel it, do proper algebra, to have an easier function to substitute.

Very easy rules for some occasions.

If the degree of the numerator is greater than the denominator.




If the degree of the numerator is equal to the denominator

If the degree of the numerator is smaller than the denominator

Domains and Ranges

Domain: are all numbers in "X" 

Range: Are all numbers in "Y" 

This table will help you to recognize the type of the domain and the range in different functions:

Here are some tips that are very useful.

If the degree of P(X) is equal to the degree on Q(x)

If the degree of P(X) is lower to the degree on Q(x)

If the degree of P(X) is lower to the degree on Q(x)

The rationals is need to know one simple a rule,the denominator should never be  0. (Domain)

The radicals also have a rule, it should never give you a negative number inside the root.

To have a range, you need to get the (X).

The rationals is need to know one simple a rule,the denominator should never be  0. (Range), exactly the same rule of Domain.

To locate a domain and a range in a table.

The domain values ​​are the farthest from the(X) axis. The range values are the farthest to the (Y) axis. 

In case if you get confused.

Functions

Definition: A function is an equation that for each value of "x" it has a value "Y".

Now a visual example:

Our function in (X)

We can assign a value to (X)

If we want our value of (X) to be "3", All we need to do is replace the (X) given in the original function.

 And you only need to solve the equation as normally you do.

Solve the necessary algebraic operations.

Done :D

Another Example, let's evaluate (X) in “1/2” .

See, So easy :D

 Another example that looks complicated but is very easy is the next one:

This is exactly the same, but now is a binomial. 

Now we have two functions:

*Caution: LOOK CAREFULLY THE SIGNS*

This small symbol tells us that the values in ​​(X) are replaced by the values ​​of G (X), but the rest of the process is the same.

Formulary

This is the formulary that we will use in Differential Calculus

Note: The little symbol that appears means "Derivative"