Using the limit definition for a derivative, I am trying to solve for the derivative of (4) / (X^.5)


the REAL answer is (-2)/(x^(3/2))

But I ended up with (-4)/(x^(3/2))

Here are my steps, as best I can write them on in text box.

so starting out we have (4 / ((x + h) ^ (.5)) - 4 / ((x)^(.5))) / h

ignoring the h on the very bottom for a while,

multiply the left side by (( x + h )^(.5) * (x ^ .5)) / ((x + h)^(.5) * (x ^ .5)) and the right side by ((x + h) ^(.5) (x + h) ^ (.5)) / ((x + h)^.5 * (x + h)^.5) (so that the denominator can eventually end up as a 3/2 power, and that the denominator is even)

Combining the now similar denominators, we get:

(sorry about mixing up how I write things, I'm just trying to write this quickly)

(4 * (X ^ .5) * ((X + h) ^ .5) - (4 * ((X + h)^.5) * ((X + h)^.5))
______________________________________…
(X^.5)((X + h)^.5)((X + h)^.5)

all over h.

simplify the numerator, we use the distributive property to pull out 5, then we multiply the two equal square roots on the right side.

4 * (X^.5((X + h)^.5) - X - h)

Split up that whole X we have into X^.5 * X^.5, then we use the distributive property on the two terms to get:

4 * (X^.5((X + h)^.5 - X^.5) - h)

Replace the leftmost h with 0, we can cancel out that entire section, because it would leave us with:

4 * (X^.5 * ((X + 0)^.5 - (X^.5)) - h) = 4 * (0 - h) = -4h
Now that we have reduced the numerator, we can cancel the very bottom h out with the one left on the top, leaving us with:

-4 / (X - h)^.5*(X - h)^.5*(X)^.5

and by replacing the remaining 2 h's with 0, we get

-4 / (X^.5)(X^.5)(X^.5) = (-4) / (X^ (3/2))

which is 2 times what the real answer is supposed to be.