How to do Derivatives

1. Let’s Do Math How to do Derivatives By NikoSurace For kids in Calculus Subject: Mathematics Go to index

2. What is a derivative • Why it is Important • Notations of Derivatives • Differentiation (Difference Quotient) • Constant and Power Rule • Sum/Difference Rule • Product Rule • Quotient Rule • Chain Rule • Trigonometric Functions • Quiz Your Self Click on a link to go to a page to learn about derivatives and then take the quiz Index of Pages

3. A derivative is in a branch of Mathematics called Calculus. The derivative is the measure of how a function changes as its input changes. It tells what the slope is at any point on a graph as well. It is one of the two properties of single variable calculus. • Back to Index What is a derivative

4. The importance of the derivatives is vital for the world today. Derivatives can tell such things like how fast some one was going, the acceleration they were experiencing, and where they were with just one equation. It also gave us the a way to prove all of the volume equations in the world. • Back to Index Why it is Important

5. There many ways to see the same notation for derivatives. Great mathematical minds used different ways to say the same thing. In class we will use Leibniz notation. • This means that dy/dx is the notation for a derivative • For multiple derviatives there is a power to d so for example the second derivative would be written as d2y/dx2 • Back to Index Notations of Derivatives

6. The difference quotient is what is formally used to find a derivative. The difference quotient is a math equation that finds the limit as x approaches a certain x value. The difference quotient is shown below. • An example of difference quotient • Back to Index Difference Quotient

7. The constant rule is used as an informal way to find the derivative. The constant rule says that any constant number when taking the derivative of it equals 0 • For instance the derivative of 5 = 0 • The power rule is a short cut to find derivatives quick and easy. The rule states that you take the exponent of any number and multiply it by the number and subtract one from the exponent itself. • Some examples • Back to Index Constant and Power Rule

8. The sum and difference rule are another informal way of finding a derivative. The sum and difference rule states that you can break up any equation by a plus or minus sign to find the derivative. • For instance x2+5 could be broken up to be d/dx x2 + d/dx 5 • Back to Index Sum/Difference Rule

9. The Product Rule is another informal way to do derivatives. The product rule as the name implies is used when taking the derivative of two things that are being multiplied together • For instance find the d/dx of xy • The product rule is f(x)gI(x) + g(x)fI(x) • An Example • Back to Index Product Rule

10. The quotient rule is also an informal way to find the derivative. It is used when to functions are divided together • For instance (x2 + 5)/(x+4) is two functions that would be perfect for the quotient rule • The quotient rule is (g(x)fI(x) – f(x)gI(x)) / (g(x))2 • Back to Index Quotient Rule

11. The chain rule is the Golden Rule. It is the rule to rule all rules. The chain rule states that for any function inside another function you take the derivative of the inside function and multiply it by the derivative of the outside function • For instance find d/dx of (x2 + 5x2 )2 • The chain rule is • Back to Index Chain Rule

12. You can take the derivative of trigonometric functions. They are always continuous and follow a pattern. This the pattern that you follow for trig functions • d/dx sin(x)= cos(x) • d/dx cos(x)= -sin(x) • d/dx tan(x)= sec2 (x) • d/dx csc(x)= -csc(x)cot(x) • d/dx sec(x)= sec(x)tan(x) • d/dx cot(x)= -csc2(x) • Back to Index Trigonometric Functions

13. Click here to begin Quiz • Back to Index Let’s Take a Quiz

14. 1) Find d/dx of x2 + 7 • A. d/dx = 2x +7 • B. d/dx = 2x • C. d/dx = x2 • D. d/dx = x2 + 7 Question 1 Back to begin quiz

15. is not the right answer • Remember to use your rules and check you math and take your time • Back to Question Answer A

16. is the correct answer • When you take the derivative of x2 using your power rule you get 2x and since 7 is a constant its derivative is 0 so your answer is 2x • Next Question Answer B

17. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer C

18. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer D

19. Find d/dx of (x2 +5)3 • A. d/dx = x6 + 125 • B. d/dx = 6(x2 + 5)3 • C. d/dx = 6x(x2 + 5)2 • D. d/dx = 3(x2 +5)2 Question 2

20. is not the right answer • Remember to use your rules and check you math and take your time • Back to Question Answer A

21. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer B

22. is the correct answer • When you take the derivative of (x2 +5)3 you have to use the chain rule and power rule functions. When you take the derivative of the inside you get 2x. You have to multiply it by the derivative of the outside function which is 3(x2 +5)2 which you should get with your power rule. This gives you the answer of 6x(x2 +5)2 • Next Question Answer C

23. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer D

24. Find d/dx of (x+5)(x+6) • A. x2 + 11x +30 • B. 2x + 11 • C. (x+5) (x+6) • D. 1 Question 3

25. is not the right answer • Remember to use your rules and check you math and take your time • Back to Question Answer A

26. is the correct answer • When you take the derivative of (x +5)(x+6) you have to use the product rule to figure it out. You have to take the derivative of both functions multiply them by the other function and then add them together. • 1(x+5) + 1(x+6) = 2x +11 • Next Question Answer B

27. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer C

28. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer D

29. Find d/dx of (x+7)/(x+3) • A. 2x+10 • B. ((x+7)(x+3))/(x-3)2 • C. (x+3)/(x+7) • D. 4/(x+3)2 Question 4

30. is not the right answer • Remember to use your rules and check you math and take your time • Back to Question Answer A

31. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer B

32. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer C

33. is the correct answer • When you take the derivative of (x +7)/(x+3) you have to use the quotient rule to figure it out. You have to take the derivative of both functions multiply them by the other function and then then subtract them by the rule and divide the differnce by the bottom squared • (1(x+7) - 1(x+3))/ (x+3)2 = 4/(x+3)2 • Next Question Answer D

34. Find d/dx of sin(6x) + 5 • A. 6cos(6x) • B. 6sin(6x) + 5 • C. 6cos(6x) + 5 • D. 6sin(6x) Question 5 The last one

35. is the correct answer • When you take the derivative of a trig function you have to use the chain rule and remember the derivative for each • Sin(6x) = 6 * cos(6x) • End Quiz Answer A

36. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer B

37. is not the right answer • Remember to use your rules and check your math and take your time • Back to Question Answer C

38. is not the right answer • Remember to use your rules and check you math and take your time • Back to Question Answer D

39. If you have any questions come and ask me. I will help you with anything I can. If anything is unclear don’t be afraid to ask me and I will explain it better. • Click here when done This is how to do Derivatives