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Home Differentiation Inverse Functions Theorem 1 Proof of Theorem 1
 
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Proof of Theorem 1

Here is a longer but complete proof which shows that dy/dx exists and computes its value.

PROOF OF THEOREM 1

Let x ≠ 0 and let Δx be nonzero infinitesimal. We first show that

Δy = (x + Δx)1/n - x1/n

is a nonzero infinitesimal. Δy ≠ 0 because x + Δx ≠ x. The standard part of Δy is

st(Δy) = st((x + Δx)1/n) - st(x1/n) = x1/n - x1/n = 0.

Therefore Δy is nonzero infinitesimal. Now

02_differentiation-237.gif

02_differentiation-238.gif

02_differentiation-239.gif

Figure 2.4.7

Figure 2.4.7 shows the graphs of y = x1/3 and y = x1/4. At x = 0, the curves are vertical and have no slope.
Home Differentiation Inverse Functions Theorem 1 Proof of Theorem 1

Last Update: 2010-11-25