Find derivative using the definition.?

Question

I need to find the derivative to this problem, f(x)=1/(2x), using the definition. I found it using the short cuts and it is -2/(2x)^2. When I work through it I get 2x though. Any idea what I'm doing wrong?

I need to find the derivative using the form, (f(x+h)-f(x))/h. I'm getting to 2/h(.5(x+h)^-1 + .5x^-1). I'm not even sure if that is right.

Answer 1

Possibly you had a mistake in simplifying the compound fraction, as it's pretty messy.

definition of the derivative is
lim h->0 [f(x+h)-f(x)]/h

Apply that to 1/2x

d[2x]/dx = lim h -> 0 [1/(2(x+h)) - 1/(2x)]/h
lim h->0 [1/(2x+2h) - 1/(2x)]/h

If you try to evaluate the limit you get 0/0, indicating a common factor of h. So simplify some to get rid of the compound fraction by multiplying top and bottom by (2x+2h)(2x) which will get rid of the denominators in the top part. Don't forget to multiply the denominator as well

lim h->0 [(2x+2h)(2x)/(2x+2h) - (2x+2h)(2x)/(2x)]/ [(h)(2x+2h)(2x)]

lim h->0 [2x-(2x+2h)]/[(h)(2x+2h)(2x)]
lim h->0 [2x-2x-2h]/[(h)(2x+2h)(2x)]

2x-2x is 0 leaving -2h on th top
lim h->0 -2h/[(h)(2x+2h)(2x)]

the h now cancels
lim h->0 -2/[(2x+2h)(2x)]

Can evaluate the limit nowAs h goes to 0 2h goes to zero
-2/[(2x)(2x)]
-2/(2x)^2

Answer 2

f(x) = 1/2x

The rule for derivative of a quotient is

d/dx u/v = ( v(du/dx) - u(dv/dx) )/ v^2

In this case, u = 1, so it's simpler
d/dx (1/v) = -(dv/dx) / v^2

v = 2x
dv/dx = 2

d/dx (1/2x) = -(2)/ (2x)^2

= -2/ (2x)^2

If you were to show us your working, we could point out the error.

Answer 3

d/dx [ x / (x + a million)] = lim as h --> 0 [ (x + h) / (x + a million + h) -- x / ( x + a million) ] / h = lim h --> 0 [{(x + h)(x + a million) -- x(x + a million + h) } / (x + a million)(x a million + h)] / h = lim h --> 0 [(x^2 + xh + x + h -- x^2 -- x -- xh) / (x + a million)(x + a million + h)] / h = lim h --> 0 [ a million / (x + a million)(x + a million + h)] = a million / (x + a million)^2 answer

Answer 4

well, maybe you're doing the power wrong, remember, it's f(x) = 0.5 x^-1, not f(x) = 2 x^-1

Answer 5

f(x+h)-f(x)
= 1/(2x+2h) -1/2x
= [2x/(2x+2h)-(2x+2h)]/[2x(2x+2h)]
= -2h/[2x(2x+2h]
Now divide by h getting
-2/[2x(2x+2h)
Now let h -->0 getting
-2/4x^2