Page 1 Higher Maths 1 3 Differentiation 1 Page 2 Higher Maths 1 3 Differentiation 1 Higher Maths 1 3 Differentiation The History of Differentiation Differentiation is part of the science of Calculus, and was first developed in the 17 th century by two different Mathematicians. Gottfried Leibniz (1646-1716) Germany Sir Isaac Newton (1642-1727) England 2 Differentiation, or finding the instantaneous rate of change, is an essential part of: • Mathematics and Physics • Chemistry • Biology • Computer Science • Engineering • Navigation and Astronomy Page 3 Higher Maths 1 3 Differentiation 1 Higher Maths 1 3 Differentiation The History of Differentiation Differentiation is part of the science of Calculus, and was first developed in the 17 th century by two different Mathematicians. Gottfried Leibniz (1646-1716) Germany Sir Isaac Newton (1642-1727) England 2 Differentiation, or finding the instantaneous rate of change, is an essential part of: • Mathematics and Physics • Chemistry • Biology • Computer Science • Engineering • Navigation and Astronomy Higher Maths 1 3 Differentiation Calculating Speed 2 4 6 8 4 8 2 6 10 0 0 Time (seconds) Distance (m) D S T × ÷ ÷ Example Calculate the speed for each section of the journey opposite. A B C speed in A = 4 3 speed in B = 5 1 5 m/s = speed in C = 2 5 0.4 m/s = average speed = 9 1.22 m/s ˜ 11 ˜ 1.33 m/s Notice the following things: • the speed at each instant is not the same as the average • speed is the same as gradient D T S = y x = m = 3 Page 4 Higher Maths 1 3 Differentiation 1 Higher Maths 1 3 Differentiation The History of Differentiation Differentiation is part of the science of Calculus, and was first developed in the 17 th century by two different Mathematicians. Gottfried Leibniz (1646-1716) Germany Sir Isaac Newton (1642-1727) England 2 Differentiation, or finding the instantaneous rate of change, is an essential part of: • Mathematics and Physics • Chemistry • Biology • Computer Science • Engineering • Navigation and Astronomy Higher Maths 1 3 Differentiation Calculating Speed 2 4 6 8 4 8 2 6 10 0 0 Time (seconds) Distance (m) D S T × ÷ ÷ Example Calculate the speed for each section of the journey opposite. A B C speed in A = 4 3 speed in B = 5 1 5 m/s = speed in C = 2 5 0.4 m/s = average speed = 9 1.22 m/s ˜ 11 ˜ 1.33 m/s Notice the following things: • the speed at each instant is not the same as the average • speed is the same as gradient D T S = y x = m = 3 Instantaneous Speed Higher Maths 1 3 Differentiation Time (seconds) Distance (m) Time (seconds) Distance (m) In reality speed does not often change instantly. The graph on the right is more realistic as it shows a gradually changing curve. The journey has the same average speed, but the instantaneous speed is different at each point because the gradient of the curve is constantly changing. How can we find the instantaneous speed? D T S = y x = m = 4 Page 5 Higher Maths 1 3 Differentiation 1 Higher Maths 1 3 Differentiation The History of Differentiation Differentiation is part of the science of Calculus, and was first developed in the 17 th century by two different Mathematicians. Gottfried Leibniz (1646-1716) Germany Sir Isaac Newton (1642-1727) England 2 Differentiation, or finding the instantaneous rate of change, is an essential part of: • Mathematics and Physics • Chemistry • Biology • Computer Science • Engineering • Navigation and Astronomy Higher Maths 1 3 Differentiation Calculating Speed 2 4 6 8 4 8 2 6 10 0 0 Time (seconds) Distance (m) D S T × ÷ ÷ Example Calculate the speed for each section of the journey opposite. A B C speed in A = 4 3 speed in B = 5 1 5 m/s = speed in C = 2 5 0.4 m/s = average speed = 9 1.22 m/s ˜ 11 ˜ 1.33 m/s Notice the following things: • the speed at each instant is not the same as the average • speed is the same as gradient D T S = y x = m = 3 Instantaneous Speed Higher Maths 1 3 Differentiation Time (seconds) Distance (m) Time (seconds) Distance (m) In reality speed does not often change instantly. The graph on the right is more realistic as it shows a gradually changing curve. The journey has the same average speed, but the instantaneous speed is different at each point because the gradient of the curve is constantly changing. How can we find the instantaneous speed? D T S = y x = m = 4 Introduction to Differentiation Higher Maths 1 3 Differentiation Differentiate means D T speed = ‘rate of change of distance with respect to time’ S T acceleration = ‘find out how fast something is changing in comparison with something else at any one instant’. gradient = y x ‘rate of change of speed with respect to time’ ‘rate of change of -coordinate with respect to -coordinate’ y x 5Read More

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