Question

Computer the derivative of f(x)=x-(1/x) for x>0 using the definition of the derivative.

Answer 1

f(x+h) = x+h - 1/(x+h)
f(x) = x - 1/x
-------------------subtract

h - 1/(x+h) + 1/x

divide by h

1 - 1/(xh+h^2) + 1/xh

1 +[ - xh +xh+h^2]/(x^2h^2+xh^3)

1 + h^2/(x^2 h^2 + x h^3)

1 + 1/(x^2 + xh)

let h --->0

1 + 1/x^2

Answer 2

How did that h on the left right below "subtract" come from??

Answer 3

uh -- from the definition of the derivative. Better reread it.

Answer 4

Yes I know the definition of the derivative, I think I'm just confused on the algebra part.

x+h - 1/(x+h)

Just that left side of the equation is where I'm lost.
It went from that to
h-1/(x+h)
Did that x just got canceled out from the left hand side? Other than that, I can understand and follow the rest of the algebra just fine

Answer 5

The definition is the limit of

f(x+h) - f(x)
-----------------
     h

That is where the subtraction comes in.

Answer 6

.. the minus made the x from the right side which forces those two to cancel out. I'm totally overlooking this problem than it needs to be. Thanks for helping me clarify a rather simple problem!