Find the derivative of the given function at the indicated point.
(use f'(a)=lim [f(a+h)-f(a)]/h as h approaches 0)
This is what I have so far:
[1/(x+h) - 1/2]/h
= [2 - (x+h)]/(h(x+h)(2))
= [2-x-h]/(h(x+h)(2))
...and now I'm stuck. Help?
Thx
oh wow. forgot the problem. the function is f(x) = 1/x, a=2
sorry haha
um...I think I solved the problem. i forgot to substitute a=2 for the first part of the function. that way when you simplify it then you get -1/4. oh dear.
To find the derivative of the given function, let's go step by step. The given function is expressed as [1/(x+h) - 1/2]/h.
Step 1: Simplify the expression
Starting with [1/(x+h) - 1/2]/h, let's work on simplifying it further.
First, let's find a common denominator for the two terms in the numerator:
1/(x+h) - (1/2) = (2/(2(x+h))) - (x+h/(2(x+h)))
= (2 - (x+h))/(2(x+h))
Now, simplify:
[(2 - x - h)/(2(x+h))]/h
= (2 - x - h)/(h * 2(x+h))
Step 2: Take the limit as h approaches 0
Now, we can apply the limit definition of the derivative. Recall that f'(a) is defined as the limit of [(f(a+h) - f(a))/h] as h approaches 0.
Applying this definition, we can take the limit as h approaches 0 of the expression we obtained in step 1:
lim[h→0] (2 - x - h)/(h * 2(x+h))
Step 3: Evaluating the limit
To evaluate the limit, we can directly substitute h = 0 into the expression and simplify:
(2 - x - 0)/(0 * 2(x+0))
= (2 - x)/(0 * 2x)
= (2 - x)/0
At this point, the expression becomes undefined because we have divided by 0.
Therefore, the derivative of the given function is undefined at the indicated point.