Comment on Activity 1

To get from the first line to the second, we multiplied the two fractions by multiplying numerators and denominators, and we used the identity

(a-b)(a+b)=a2-b2.

To get from the second line to the third, we used the fact that the squaring function is the inverse of the square root function. To get from the third line to the fourth, we simplified the numerator to Δx and canceled this Δx with the one in the denominator.

As Δx0, x+Δx+x approaches the limiting value 2x. Thus 1x+Δx+x approaches the limiting value 12x. This shows that

ddxx=12x.