Computer the derivative of f(x)=x-(1/x) for x>0 using the definition of the derivative.
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
How did that h on the left right below "subtract" come from??
uh -- from the definition of the derivative. Better reread it.
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
The definition is the limit of
f(x+h) - f(x)
-----------------
h
That is where the subtraction comes in.
.. 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!
To compute the derivative of the function f(x) = x - (1/x) using the definition of the derivative, we need to go through a few steps.
Step 1: Recall the definition of the derivative. The derivative of a function f(x) at a particular point x is given by the limit as h approaches 0 of (f(x + h) - f(x)) / h.
Step 2: Apply the definition of the derivative to f(x). We have:
f'(x) = lim(h->0) [(f(x + h) - f(x)) / h]
Step 3: Substitute the function f(x) into the definition. We have:
f'(x) = lim(h->0) [(x + h - (1/(x + h))) - (x - (1/x))] / h
Step 4: Simplify the expression inside the limit.
f'(x) = lim(h->0) [(x + h - 1/(x + h) - x + 1/x)] / h
= lim(h->0) [h - (1/(x + h) - 1/x)] / h
Step 5: Expand the expression and combine like terms.
f'(x) = lim(h->0) [h - (x - x - h)/(x(x + h))] / h
= lim(h->0) [h - (-h)/(x(x + h))] / h
= lim(h->0) [h + h / (x(x + h))] / h
Step 6: Cancel out the h terms.
f'(x) = lim(h->0) [1 + 1 / (x(x + h))]
Step 7: Take the limit as h approaches 0.
f'(x) = 1 + 1 / (x^2)
So, the derivative of f(x) = x - (1/x) is f'(x) = 1 + 1 / (x^2).