Introduction to derivates (laws and properties).pptx

Derivatives are a fundamental concept in calculus used to measure how a function changes as its input changes. In simple terms, the derivative represents the rate of change or the slope of a curve at a specific point.

They are widely used in science and engineering to analyze motion, optimize systems, and model real-world situations. Common rules include the power rule, product rule, quotient rule, and chain rule, which simplify the process of differentiation.

Understanding derivatives helps describe how quantities vary over time — for example, how velocity is the derivative of position, and acceleration is the derivative of velocity.

1. U3 Derivatives

2. What is derivatives? It is what allows us to find the tangent line to a given function .   NOTE: is the Increment of X  

3. Slope of a line If two random points P 1 (x 1 ,y 1 ) and P 2 (x 2 ,y 2 ) are chosen on a straight line , the ratio is being this a constant and is called the slope of the straight line and is defined as: Ejemplo :                     (1,3) y (3,7)       7-3 3-1   The equation of the straight line in its slope ordered form is:   1  

4. Secant line A secant line is any line that cuts a curve at 2 points . Tangent Line The tangent is any line that contains one and only one point of the curve, which is called the tangent point.

5. Average rate of change   1.- Graphic 2.- Type of function     3.- Mathematical function   4.       5. Average speed 3hrs=     5hrs=    

6. Instantaneous velocity                            

7. Secant line       Slope of the secant line

8. Tangent Line         Slope of the tangent line or instantaneous velocity

9. Formal definition of the derivative Let be a function and its derivative is defined as:         All provided that the limit exists and is represented by:   o    

10. Take the derivative of     Example 1 Step 1. Know the value of =3 ( ) +2 Step 2. Take the value of   Tip:  

11.   Example 2 Step 1. Know the value of Step 2. Take the value of   Take the derivative of  

12. How to factor Step 1. First in each of the terms must be the . Step 2. You have to divide between .      

13. How to factor Step 1. First in each of the terms must be the . Step 2. You have to divide between .      

14.         NOTA: In order to divide between     Step 1. Arrange the values in the formula Step 2. Make the respective multiplications

15.         Step 3. Removing terms Step 4. Solve the limit =3   Paso 5. Since in this case there is no , IT CANNOT BE REPLACED BY 0.   So the result is  

16.     Exercises to solve 1 2   3   4   5   6 1  

17. LIST OF DERIVATIVES

18. Derivation rules If then the derivative is Example : then the derivative is   If then the derivative is Example : then the derivative is   in Square Root Example : then the derivative is  

19. Derivation rules If then the derivative is Example : then the derivative is   if then the derivative is Ejemplo: then the derivative is  

20. Derivation rules If then the derivative is Example : then the derivative is   If then the derivative is Example : then the derivative is  

21. Rules of the chain Step 1. Derived from outside and derived from inside Step 2. Do the operations  

22. Derivation rules If then the derivative is Example : then the derivative is   If then the derivative is Example : then the derivative is  

23. Derivation rules If then the derivative is Example : then the derivative is But by the chain rule the angle can always be derived, we derive it   If then the derivative is Example : then the derivative is But by the chain rule the angle can always be derived, we derive it  

24. Derivation rules If then the derivative is Example : then the derivative is But by the chain rule the angle can always be derived, we derive it   If then the derivative is Example : then the derivative is But by the chain rule the angle can always be derived, we derive it  

25. Derivation rules If then the derivative is Example : then the derivative is But by the chain rule the angle can always be derived, we derive it   If then the derivative is Example : then the derivative is But by the chain rule the angle can always be derived, we derive it  

26. Rule of inverse trigonometric   Ejemplo:  

27. Implicit derivatives Explicit because the letter 𝑦 is the dependent variable is clear. Implicit because the variable 𝑦 which is the dependent variable is no longer clear, but is found within the equation.  

28. Example of implicit derivatives: Differentiating with respect to: Exercises to practice :   Implicit derivatives