Comment on Activity 2

For this derivative, the difference quotient is

`(text[(]x+Delta xtext[)]sqrt(x+Delta x)-xsqrt(x))/(Delta x)`.

If we multiply both numerator and denominator by `text[(]x+Delta xtext[)]sqrt(x+Delta x)+xsqrt(x)`, we have

`([text[(]x+Delta xtext[)]sqrt(x+Delta x)]^2-text[(]xsqrt(x)text[)]^2)/(Delta x[text[(]x+Delta xtext[)]sqrt(x+Delta x)+xsqrt(x)])`.

This simplifies to

`(text[(]x+Delta xtext[)]^3-x^3)/(Delta x[text[(]x+Delta xtext[)]sqrt(x+Delta x)+xsqrt(x)])`.

When we expand the numerator and cancel, we find

`(3x^2Delta x+3xtext[(]Delta xtext[)]^2+text[(]Delta xtext[)]^3)/(Delta x[text[(]x+Delta xtext[)]sqrt(x+Delta x)+xsqrt(x)])`.

Next we cancel the `Delta x` in the denominator with a factor of `Delta x` in the numerator to obtain

`(3x^2+3xDelta x+text[(]Delta xtext[)]^2)/(text[(]x+Delta xtext[)]sqrt(x+Delta x)+xsqrt(x))`.

Now the numerator of this approaches `3x^2` as `Delta x->0`, and the denominator approaches `xsqrt(x)+xsqrt(x)=2xsqrt(x)`. Thus the limiting value of the difference quotient is `text[(]3x^2text[)/(]2xsqrt(x)text[)]` or `text[(]3sqrt(x)text[)/]2`.

If we use the Product Rule and the formula for the derivative of the square root function, we find

`d/(dx)xsqrt(x)=1*sqrt(x)+x*1/(2sqrt(x))=3/2sqrt(x)`.

Both approaches give the same formula for the derivative, but the second approach is certainly easier.