121616949-math.71 - Δ x → involves what happens when Δ...

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3.5 The Chain Rule 57 EXAMPLE 3.8 Form the two possible compositions of f ( x ) = x and g ( x ) = 625 - x 2 . First, f ( g ( x )) = 625 - x 2 ; second g ( f ( x )) = 625 - ( x ) 2 = 625 - x . Suppose we want the derivative of f ( g ( x )). Again, let’s set up the derivative and play some algebraic tricks: d dx f ( g ( x )) = lim Δ x 0 f ( g ( x + Δ x )) - f ( g ( x )) Δ x = lim Δ x 0 f ( g ( x + Δ x )) - f ( g ( x )) g ( x + Δ x )) - g ( x ) g ( x + Δ x )) - g ( x ) Δ x Now we see immediately that the second fraction turns into g ( x ) when we take the limit. The first fraction is more complicated, but it too looks something like a derivative. The denominator, g ( x + Δ x )) - g ( x ), is a change in the value of g , so let’s abbreviate it as Δ g = g ( x + Δ x )) - g ( x ), which also means g ( x + Δ x ) = g ( x ) + Δ g . This gives us lim Δ x 0 f ( g ( x ) + Δ g ) - f ( g ( x )) Δ g . As Δ x goes to 0, it is also true that Δ g goes to 0, because g ( x + Δ x ) goes to g ( x ). So we can rewrite this limit as lim Δ g 0 f ( g ( x ) + Δ g ) - f ( g ( x )) Δ g . Now this looks exactly like a derivative, namely f ( g ( x )), that is, the function f ( x ) with x replaced by g ( x ). If this all withstands scrutiny, we then get d dx f ( g ( x )) = f ( g ( x )) g ( x ) . Unfortunately, there is a small flaw in the argument. Recall that what we mean by lim
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Unformatted text preview: Δ x → involves what happens when Δ x is close to 0 but not equal to 0. The qualification is very important, since we must be able to divide by Δ x . But when Δ x is close to 0 but not equal to 0, Δ g = g ( x + Δ x ))-g ( x ) is close to 0 but possibly equal to 0. This means it doesn’t really make sense to divide by Δ g . Fortunately, it is possible to recast the argument to avoid this difficulty, but it is a bit tricky; we will not include the details, which can be found in many calculus books. The chain rule has a particularly simple expression if we use the Leibniz notation for the derivative. The quantity f ± ( g ( x )) is the derivative of f with x replaced by g ; this can be written df/dg . As usual, g ± ( x ) = dg/dx . Then the chain rule becomes df dx = df dg dg dx ....
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