The Derivative by Definition
Key Questions

First you have to calculate the derivative of the function.
#f(x)=x^3# #f'(x)=3x^2# Then if we want to find the derivative of
#f(x)# when#x=4# then we substitute that value into#f'(x)# .#f'(4)=3(4)^2=3*16=48# 
The formal definition of derivative of a function
#y=f(x)# is:
#y'=lim_(Deltax>0)(f(x+Deltax)f(x))/(Deltax)# The meaning of this is best understood observing the following diagram:
The secant PQ represents the mean rate of change
#(Deltay)/(Deltax)# of your function in the interval between#x# and#x+Deltax# .If you want the rate of change, say, at P you "move" point Q (and the secant with it) to meet point P as in:
In doing so you must reduce
#Deltax# . If#Delta x>0# you'll get the tangent in P whose inclination will give the inclination at P and thus the derivative at P.Hope it helps!

Answer:
We use quotient rule as described below to differentiate algebraic fractions or any other function written as quotient or fraction of two functions or expressions
Explanation:
When we are given a fraction say
#f(x)=(32xx^2)/(x^21)# . This comprises of two fractions  say one#g(x)=32xx^2# in numerator and the other#h(x)=x^21# , in the denominator. Here we use quotient rule as described below.Quotient rule states if
#f(x)=(g(x))/(h(x))# then
#(df)/(dx)=((dg)/(dx)xxh(x)(dh)/(dx)xxg(x))/(h(x))^2# Here
#g(x)=32xx^2# and hence#(dg)/(dx)=22x# and as#h(x)=x^21# , we have#(dh)/(dx)=2x# and hence#(df)/(dx)=((22x)xx(x^21)2x xx(32xx^2))/(x^21)^2# =
#(2x^32x^2+2x+26x+4x^2+2x^3)/(x^21)^2# =
#(2x^24x+2)/(x^21)^2# or
#(2(x1)^2)/(x^21)^2# =
#2/(x+1)^2# Observe that
#(32xx^2)/(x^21)=((1x)(3+x))/((x+1)(x1))=(3x)/(x+1)# and using quotient rule#(df)/(dx)=((x+1)(3x))/(x+1)^2=2/(x+1)^2#