# Use the four step process to find the derivative of f(x) where f'(x) = lim [f(x+h)-f(x)]/h (the lim is h to 0) : f(x) = 1 / 4x-3

Using the 4 step process you can find f'(x) below:

1) f(x+h) = 1/ (4(x+h)-3) and f(x) = 1 / 4x-3

2) f(x+h) - f(x)= (4x - 3)/(4x + 4h - 3)(4x - 3) - (4x + 4h - 3)/(4x + 4h - 3)(4x - 3) = [ (4x-3)-(4x + 4h -3)] / (4x + 4h - 3)(4x - 3) = - 4h / 16x^2 + 16xh - 24x - 12h + 9

3) f(x+h) - f(x) / h = [- 4h /16x^2 + 16xh - 24x - 12h +9] / h = [- 4h /16x^2 + 16xh - 24x - 12h +9] * (1/h) = (cancel h's) - 4 /16x^2 + 16xh - 24x - 12h +9

4) as h --> 0 using step 3

f'(x) = lim f(x+h) - f(x) / h =

lim - 4 /16x^2 + 16xh - 24x - 12h +9 {as h --> 0} = - 4 /16x^2 + 0 - 24x - 0 +9 = - 4 / 16x^2 - 24x + 9

So f'(x) = - 4 / (16x^2 - 24x + 9).

**** the quotient rule is much easier to use here, but as you can see, you should always get the same answer regardless of the method used***

f(x) = 1/4x-3

To find the derivative of f(x) = f'(x) defined like Lim h-->0 (f(x+h)-f(x))/h.

Therefore,

f'(x) = lim h-->0 [[1/4(x+h)-3 ]-[1/4x-3]}

=lim h-->0 [1/4(x+h)-1/4x]/h

=lim h-->0 [x-(x+h)]/[4(x+h)(x)h],

lim h-->0 [-h]/[4(x+h)(x)h],h gets cancelled in numerator and denominator.

=Lim h-->0 [-1]/(4x+h)(x) = -1/[4x+0)x] = 1/94x^2)

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