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This document is a presentation submitted by a group of 6 mechanical engineering students to their professor. It contains an introduction, definitions of derivatives, a brief history of derivatives attributed to Newton and Leibniz, and applications of derivatives in various fields such as automobiles, radar guns, business, physics, biology, chemistry, and mathematics. It also provides rules and examples of calculating derivatives using power, multiplication by constant, sum, difference, product, quotient and chain rules.

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derivatives math
PRESENTATION ON DERIVATION
SUBMITTED TO: Mam SADIA FIRDUS SUBMITTED BY: GROUP NO. 03
BS-MECHANICAL TECHNOLOGY (1ST SEMESTER) NAME ROLL NO. 1. AHSAN WAQAR 17093386-032 2. ALI HASSAN 17093386-015 3. ALI RAZA 17093386-031 4. USMAN ZAFAR 17093386-018 5. ABDULLAH 17093386-035 6. Muhammad Umer 17093386-036
Contents Definition of derivatives . History of derivatives. Application of derivatives. Derivatives rules & examples.
DEFINATION OF DERIVATION? 1. The derivatives is the exact rate at which one quantity changes with respect to another. 2. Geometrically, the derivatives is the slope of curve at point on curve. 3. The derivatives is often called the instantaneous rate of change. 4. The derivatives of a function represents an infinitely small change the fuction with respect to one of its variable. Its is written as dy dx
NOTE: does not mean dy dx (except when it is convenient to think of it as division.) df dx does not mean df dx (except when it is convenient to think of it as division.)
Note: dx does not mean d times x ! dy does not mean d times y !
History of the derivation
SIR ISAAC NEWTON
Sir leibniz
derivatives math
APPLICATIONS OF DERIVATIVES in real life.
Automobiles In an automobile there is always an odometer and a speedometer. These two gauges work in tandem and allow the driver to determine his speed and his distance that he has traveled. Electronic versions of these gauges simply use derivatives to transform the data sent to the electronic motherboard from the tires to miles per Hour(MPH) and distance(KM).
Radar Guns Keeping with the automobile theme from the previous slide , all police officers who use radar guns are actually taking advantage of the easy use of derivatives. When a radar gun is pointed and fired at your care on the highway. The gun is able to determine the time and distance at which the radar was able to hit a certain section of your vehicle. With the use of derivative it is able to calculate the speed at which the car was going and also report the distance that the car was from the radar gun.
Business In the business world there are many applications for derivatives. One of the most important application is when the data has been charted on graph or data table such as excel. Once it has been input, the data can be graphed and with the applications of derivatives you can estimate the profit and loss point for certain ventures.
Applications of Derivatives in Various fields/Sciences: Such as in: Physics Biology Economics Chemistry Mathematics
Derivatives in Physics: In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity W.R.T time is acceleration. Newtons second law of motion states that the derivative of the momentum of a body equals the force applied to the body.
Derivatives in Biology: The instantaneous rate of change does not make exact sense in the previous example because the change in population is not exactly a continuous process. However, for large population we can approximate the population function by a smooth(continuous) curve. Example: Suppose that a population of bacteria doubles its population , n, every hour. Denote by n0 the initial population i.e. n(0)=n0. In general then, n(t)=2t no Thus the rate of growth of the population at time t is (dn/dt)=no2tln2
Derivatives in Chemistry: One use of derivatives in chemistry is when you want to find the concentration of an element in a product. Derivative is used to calculate rate of reaction and compressibility in chemistry
Derivatives in Mathematics: The most common use of the derivatives in Mathematics is to study functions such as: Extreme values of function The Mean Value theorem Monotonic functions Concavity & curve sketching Newtons Method etc.
Rules & example of derivation: Rules of derivative are as under:
Power Rule: Example: What is 3 ? The question is asking "what is the derivative of 3 ?" We can use the Power Rule, where n=3: = nxn1 3 = 3 31 = 3 2 (In other words the derivative of 3is32)
Multiplication by constant: Example: What is 5 3 ? the derivative of cf = cf the derivative of 5f = 5f We know (from the Power Rule): 3 = 3 31=3 2 So: 53 = 5 3 = 5 32 = 15 2
Sum Rule: Example: What is the derivative of 2+3 ? The Sum Rule says: the derivative of f + g = f + g So we can work out each derivative separately and then add them. Using the Power Rule: 2 = 2x 3 = 3 2 And so: the derivative of 2+ 3 = 2x + 32
Difference Rule: Example: What is ( 3 4 ) ? The Difference Rule says the derivative of f g = f g So we can work out each derivative separately and then subtract them. Using the Power Rule: 3 = 3 2 4 = 4 3 And so: the derivative of 3 4 = 3243
Product Rule: Example: What is the derivative of cos(x)sin(x) ? The Product Rule says: the derivative of fg = f g + f g In our case: f = cos g = sin We know (from the table above): cos(x) = sin(x) sin(x) = cos(x) So: the derivative of cos(x)sin(x) = cos(x)cos(x) sin(x)sin(x) = 2(x) 2(x)
Quotient rule: Examples: The quotient rule can be used to find the derivative of As follow:
Chain rule: Example: What is sin(x2) ? sin(x2) is made up of sin() and x2: f(g) = sin(g) g(x) = x2 The Chain Rule says: the derivative of f(g(x)) = f'(g(x))g'(x) The individual derivatives are: f'(g) = cos(g) g'(x) = 2x So: sin(x2) = cos(g(x)) (2x) = 2x cos(x2)
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derivatives math

  • 3. SUBMITTED TO: Mam SADIA FIRDUS SUBMITTED BY: GROUP NO. 03
  • 4. BS-MECHANICAL TECHNOLOGY (1ST SEMESTER) NAME ROLL NO. 1. AHSAN WAQAR 17093386-032 2. ALI HASSAN 17093386-015 3. ALI RAZA 17093386-031 4. USMAN ZAFAR 17093386-018 5. ABDULLAH 17093386-035 6. Muhammad Umer 17093386-036
  • 5. Contents Definition of derivatives . History of derivatives. Application of derivatives. Derivatives rules & examples.
  • 6. DEFINATION OF DERIVATION? 1. The derivatives is the exact rate at which one quantity changes with respect to another. 2. Geometrically, the derivatives is the slope of curve at point on curve. 3. The derivatives is often called the instantaneous rate of change. 4. The derivatives of a function represents an infinitely small change the fuction with respect to one of its variable. Its is written as dy dx
  • 7. NOTE: does not mean dy dx (except when it is convenient to think of it as division.) df dx does not mean df dx (except when it is convenient to think of it as division.)
  • 8. Note: dx does not mean d times x ! dy does not mean d times y !
  • 9. History of the derivation
  • 14. Automobiles In an automobile there is always an odometer and a speedometer. These two gauges work in tandem and allow the driver to determine his speed and his distance that he has traveled. Electronic versions of these gauges simply use derivatives to transform the data sent to the electronic motherboard from the tires to miles per Hour(MPH) and distance(KM).
  • 15. Radar Guns Keeping with the automobile theme from the previous slide , all police officers who use radar guns are actually taking advantage of the easy use of derivatives. When a radar gun is pointed and fired at your care on the highway. The gun is able to determine the time and distance at which the radar was able to hit a certain section of your vehicle. With the use of derivative it is able to calculate the speed at which the car was going and also report the distance that the car was from the radar gun.
  • 16. Business In the business world there are many applications for derivatives. One of the most important application is when the data has been charted on graph or data table such as excel. Once it has been input, the data can be graphed and with the applications of derivatives you can estimate the profit and loss point for certain ventures.
  • 17. Applications of Derivatives in Various fields/Sciences: Such as in: Physics Biology Economics Chemistry Mathematics
  • 18. Derivatives in Physics: In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity W.R.T time is acceleration. Newtons second law of motion states that the derivative of the momentum of a body equals the force applied to the body.
  • 19. Derivatives in Biology: The instantaneous rate of change does not make exact sense in the previous example because the change in population is not exactly a continuous process. However, for large population we can approximate the population function by a smooth(continuous) curve. Example: Suppose that a population of bacteria doubles its population , n, every hour. Denote by n0 the initial population i.e. n(0)=n0. In general then, n(t)=2t no Thus the rate of growth of the population at time t is (dn/dt)=no2tln2
  • 20. Derivatives in Chemistry: One use of derivatives in chemistry is when you want to find the concentration of an element in a product. Derivative is used to calculate rate of reaction and compressibility in chemistry
  • 21. Derivatives in Mathematics: The most common use of the derivatives in Mathematics is to study functions such as: Extreme values of function The Mean Value theorem Monotonic functions Concavity & curve sketching Newtons Method etc.
  • 22. Rules & example of derivation: Rules of derivative are as under:
  • 23. Power Rule: Example: What is 3 ? The question is asking "what is the derivative of 3 ?" We can use the Power Rule, where n=3: = nxn1 3 = 3 31 = 3 2 (In other words the derivative of 3is32)
  • 24. Multiplication by constant: Example: What is 5 3 ? the derivative of cf = cf the derivative of 5f = 5f We know (from the Power Rule): 3 = 3 31=3 2 So: 53 = 5 3 = 5 32 = 15 2
  • 25. Sum Rule: Example: What is the derivative of 2+3 ? The Sum Rule says: the derivative of f + g = f + g So we can work out each derivative separately and then add them. Using the Power Rule: 2 = 2x 3 = 3 2 And so: the derivative of 2+ 3 = 2x + 32
  • 26. Difference Rule: Example: What is ( 3 4 ) ? The Difference Rule says the derivative of f g = f g So we can work out each derivative separately and then subtract them. Using the Power Rule: 3 = 3 2 4 = 4 3 And so: the derivative of 3 4 = 3243
  • 27. Product Rule: Example: What is the derivative of cos(x)sin(x) ? The Product Rule says: the derivative of fg = f g + f g In our case: f = cos g = sin We know (from the table above): cos(x) = sin(x) sin(x) = cos(x) So: the derivative of cos(x)sin(x) = cos(x)cos(x) sin(x)sin(x) = 2(x) 2(x)
  • 28. Quotient rule: Examples: The quotient rule can be used to find the derivative of As follow:
  • 29. Chain rule: Example: What is sin(x2) ? sin(x2) is made up of sin() and x2: f(g) = sin(g) g(x) = x2 The Chain Rule says: the derivative of f(g(x)) = f'(g(x))g'(x) The individual derivatives are: f'(g) = cos(g) g'(x) = 2x So: sin(x2) = cos(g(x)) (2x) = 2x cos(x2)