find the derivative of g(x)=2/3x?

Question

Compute the derivate of Q(x)=2/3x, using the limit definition. just before you subsititue h=0, what expression are you taking the limit of?

Answer 1

I assume you mean 2/(3x).

g'(x) = lim(h->0) [ 2/(3(x+h)) - 2/(3x) ] / h
= lim (h->0) [ (x/x)(2/(3(x+h))) - 2/(3x) (x+h)/(x+h) ] / h
= lim (h->0) [ (2x - 2(x+h)) / (3x(x+h)) ] / h
= lim (h->0) [ (-2h) / (3x(x+h)) ] / h
= lim (h->0) -2 / (3x(x+h))

Now plug in h=0, and you get the limit, -2/(3x^2).

Answer 2

g'(x) = lim h->0 [ 2/(3(x+h)) - 2/(3x) ] / h
= lim h->0 [ {2x -2( x+h) }/3x(x+h) ] / h
= lim (h->0) [ (-2h) / (3x(x+h)) ] / h
= lim (h->0) (-2) / (3x(x+h))
= (-2) / (3x(x)) = -2/(3x^2)