Chain rule

Chain rule - y = [ g ( x )] n , we get dy dx = dy du du dx...

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CHAIN RULE Recall the bottle calibration problem. If we increase the amount of water dripped into a bottle twice as much, then, no matter what the shape of the bottle is, the height of the water will raise twice as fast. This suggests that, if we have a composite function f ( g ( x )), then the derivative of this composite function should contain g 0 ( x ) term. Here, we can think f denotes the height function, and g is the amount of water, so that the derivative of the composition means the rate of change of the height. In Leibniz notation, if y = f ( u ) and u = g ( x ) are both differentiable functions, then dy dx = dy du du dx . If we write it in the prime notation, it becomes ( f g ) 0 ( x ) = f 0 ( g ( x )) · g 0 ( x ) . It means that the derivative of the composite function f g is the product of the derivatives f and g , and this fact is called the Chain Rule . Example 1. Differentiate f ( x ) = e x . Solution f 0 ( x ) = e x · 1 2 x By using the Chain Rule and the Power Rule, if
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Unformatted text preview: y = [ g ( x )] n , we get dy dx = dy du du dx = nu n-1 du dx = n [ g ( x )] n-1 g ( x ) . Here, we wrote y = u n where u = g ( x ). Example 2. Dierentiate f ( x ) = 3 x 2 + x + 1. Solution f ( x ) = 1 3 ( x 2 + x + 1)-2 3 (2 x + 1) We can use the chain rule to dierentiate an exponential function with any base. If we use a x = ( e ln a ) x = e (ln a ) x , then d dx ( a x ) = d dx e (ln a ) x = e (ln a ) x d dx (ln a ) x = e (ln a ) x ln a = a x ln a 1 2 CHAIN RULE If we have a longer chain, we can use the chain rule twice or more. Example 3. Dierentiate f ( x ) = sin(sin(sin x )). Solution f ( x ) = cos(sin(sin x )) d dx sin(sin x ) = cos(sin(sin x )) cos(sin x ) d dx (sin x ) = cos(sin(sin x )) cos(sin x ) cos x...
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Chain rule - y = [ g ( x )] n , we get dy dx = dy du du dx...

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