Use the Limit Definition of the Derivative to find derivative of f(x)=1/sqrt(9x)

Question

Confused on how to do the Definition of A Derivative.

Answer 1

Definition: Derivative of the function y=f(x) at the point x0 is defined as

f'(x0)=lim_h→0(f(x₀+h)-f(x₀))/h. This is general definitio and x0 is arbitrary point from D(f).

Now, in you question f(x)=1/√(9x)=1/(3√x). First find f(x+h)=1/(3√(x+h)).

According to the above defintion

f'(x)=lim_h→0[1/(3√(x+h))-1/(3√x)]/h. (LIM)

Only thing (most important) left is to find the limit in RHS (LIM)

lim_h→0[1/(3√(x+h))-1/(3√x)]/h=

=lim_h→0(1/(3h))(1/√(x+h)-1/√x)=

=(1/3)lim_h→0(1/h)(1/√(x+h)-1/√x)=

=(1/3)lim_h→0(1/h)(√x-√(x+h))/√(x(x+h))=

=(1/3)lim_h→0(1/h)(x-(x+h))/[√(x(x+h))*(√x+√(x+h)]=

=-(1/3)lim_h→01/[√(x(x+h))*(√x+√(x+h)]=

=-(1/3)1/[√(x2)(2√x)]=-1/(6x√x).

So, 

f'(x)=-1/(6x√x)=1/(6x^3/2)

Check it.