Calc1231W3T2 - The Chain Rule

Saturday, 4 August 2018

11:15 AM

Machine generated alternative text: The chain rule allows you to take the derivative of the composite function For example if f@) = 5x + 4 and g@) — 5 x + 4x + 5 then Hence by the chain rule dc dc

 

Machine generated alternative text: In the previous question we used the chain rule to calculate the derivative f o g indirectly from the derivatives of f and g. Of course; in the previous question f and g were polynomials, and so a simpler method to find the derivative would be to first evaluate f o g and then differentiate. However; this "simpler" method does not always work. For example, use the chain rule dc to evaluate d dc 4 4 dc e d dc d —1 tan dc Recall (4xA3rcos(xA4+1) 4Xh3/(XA4+1) 4 —1 1

 

Machine generated alternative text: What if you are faced with taking the derivative of a function of a function of a function? A single application of the chain rule is not enough. DEEPER From Know Your Meme We need to use the chain rule within the chain rule! For example, to evaluate Let f(x) = sin@) and g(x) d. Then a first application of the chain rule gives (2 +2 x+5) d It is possible at this stage to find However to find g @) requires a further use of the chain rule Hence dc • e(2x2+2X+5))

 

Machine generated alternative text: The kinetic energy of a object with mass m (kilograms) and velocity v (metres per second) is given by Suppose you drop an orange, where • the orange has a mass of m = 5/6 kilograms, • the acceleration due to gravity is dt —9.8 metres per second per second. By the chain rule; the change in kinetic energy with respect to time is a function of v: dlC dt When the velocity is 4 metres per second, the rate of change in kinetic energy is dlC 98 -98/3

 

Machine generated alternative text: Remarkably the chain rule allows you to calculate the derivative of the composite function f o g@) = as the product of the derivatives f' (g(x)) and g'@) . It is not actually necessary to know an explicit formula for the functions f and g , only their derivatives are required. For example, suppose you drive a car up a steep mountain road at a constant speed of v = 8 kilometers per hour The gradient of the mountain is 30 degrees (l told you it was steep') If g(t) represents the altitude (in kilometers) of the car at time t (in hours) then, using the triangle below, we can calculate the rate of change in altitude as O kilometers per hour. 300 13 Now let f(h) represent the temperature at h kilometers above sea level. The temperature drops at a constant rate of degrees for every kilometer above sea level. Hence 2 The composite function represents the temperature at time t. By the chain rule the temperature changes with time at a rate of O degrees per hour. This calculation was done without knowing the value oft, h, g(t) or f(h)- Note: this question was adapted from this example on Wikipedia

 

 

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