This is the formula for computing the derivative using the limit definition.
#lim_(delta x -> 0) (f(x+delta x) - f(x)) / (delta x)#
Substituting in our function #6/x#:
#=lim_(delta x -> 0) (6/(x+delta x) - 6/x) / (delta x)#
Establishing a common denominator in the numerator:
#=lim_(delta x -> 0) (6/(x+delta x) * (x)/(x) - 6/x*(x+ delta x)/(x+ delta x)) / (delta x)#
#=lim_(delta x -> 0) ((6x)/(x(x+delta x)) - (6(x+ delta x))/(x(x+delta x))) / (delta x)#
Combining fractions with the common denominator:
#=lim_(delta x -> 0) ((6x - 6(x+ delta x))/(x(x+delta x))) / (delta x)#
Multiplying out:
#=lim_(delta x -> 0) ((6x - 6x- 6delta x)/(x^2+xdelta x)) / (delta x)#
We have #6x# which cancels the #-6x# in the numerator:
#=lim_(delta x -> 0) -((6delta x)/(x^2+xdelta x)) / (delta x)#
We have #delta x# which cancels the #delta x# in the denominator:
#=lim_(delta x -> 0) -(6)/(x^2+xdelta x)#
And we're left with:
#=-6/(x^2)#
