Remarkably, the chain rule allows you to calculate the derivative of the composite functionf?g(x)=f(g(x))

as the product of the derivativesf

?

(g(x))

andg

?

(x)

. It is not actually necessary to know an explicit formula for the functionsf

andg

, only their derivatives are required.

For example, suppose you drive a car up a steep mountain road at a constant speed ofv=8

kilometers per hour. The gradient of the mountain is30

degrees (I told you it was steep!) Ifg(t)

represents the altitude (in kilometers) of the car at timet

(in hours) then, using the triangle below, we can calculate the rate of change in altitude as

Remarkably, the chain rule allows you to calculate the derivative of the composite function f 0 9(3) 2 119(2)] as the product of the derivatives f J19(3)) and g"I [a] . It is to know an explicit formula for the functions f and 9, only their derivatives are required. For example, suppose you drive a car up a steep mountain road at a constant speed of o = 8 kilometers per hour. The gradient ofthe mountain is 30 degrees (I told you it 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 g'(t) : [Number I kilometers per hour. E Now let h) represent the temperature at h kilometers above sea level. The temperature drops at a constant rate of 2 f' (h) z |' Number '| The composite function f(g(t}) represents the temperature at time t. Ely:r the chain rule the temperature changes with time at a rate of % f[g(t)] = fr(g(t)]g" [t] = ' Number I degrees per hour. This calculation was done without knowing the value of t, h, g[t) or h). degrees for every kilometer above sea level. Henc